Volume with cross sections: triangle

WEBVTT
Kind: captions
Language: en

00:00:01.163 --> 00:00:02.648
- [Voiceover] Lets see if we can imagine

00:00:02.648 --> 00:00:05.141
a three-dimensional shape

00:00:05.141 --> 00:00:07.889
whose base could be viewed as

00:00:07.889 --> 00:00:10.688
this shaded in region between

00:00:10.688 --> 00:00:13.567
the graphs of Y is equal to F of X

00:00:13.567 --> 00:00:15.734
and Y is equal G of X.

00:00:15.734 --> 00:00:17.607
That's the base of our ...

00:00:17.607 --> 00:00:20.397
this purple, this I guess
mauve or purple color

00:00:20.397 --> 00:00:22.286
is the base of it.

00:00:22.286 --> 00:00:24.791
It's kind of popping out of our screen.

00:00:24.791 --> 00:00:26.102
What I've drawn here in blue,

00:00:26.102 --> 00:00:28.099
you could view this kind of the top ridge

00:00:28.099 --> 00:00:29.944
of the figure.

00:00:29.944 --> 00:00:32.834
If you were to take cross
sections of the figure,

00:00:32.834 --> 00:00:34.659
that's what this yellow line is.

00:00:34.659 --> 00:00:37.125
If you were to take cross
sections of this figure,

00:00:37.125 --> 00:00:39.853
that our vertical, I should say

00:00:39.853 --> 00:00:42.625
our perpendicular to the X axis,

00:00:42.625 --> 00:00:44.899
those cross sections are going to be

00:00:44.899 --> 00:00:47.518
isosceles right triangles.

00:00:47.518 --> 00:00:49.055
This cross section

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is going to look like this,

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if you were going to flatten it out.

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So over here it's sitting.

00:00:54.481 --> 00:00:57.623
It's popping out of your
page or out of your screen.

00:00:57.623 --> 00:01:00.719
If you were to actually flatten it out,

00:01:00.719 --> 00:01:03.300
the cross sections would look like this.

00:01:03.300 --> 00:01:05.738
It's going to be an
isosceles right triangle

00:01:05.738 --> 00:01:09.826
with a hypotenuse of the
isosceles right triangle

00:01:09.826 --> 00:01:12.632
sits along the base.

00:01:12.632 --> 00:01:14.315
Sits along the base.

00:01:14.315 --> 00:01:16.339
So, it's isosceles. So
that's equal to that.

00:01:16.339 --> 00:01:17.878
It's a right triangle,

00:01:17.878 --> 00:01:20.073
and then this distance

00:01:20.288 --> 00:01:23.112
this distance between that point

00:01:23.112 --> 00:01:27.205
and this point is the same as the distance

00:01:27.205 --> 00:01:30.111
between F of X and G of X.

00:01:30.449 --> 00:01:35.449
F of X and G of X for this
X value right over there.

00:01:36.147 --> 00:01:40.952
Now obviously that changes
as we change our X value.

00:01:40.952 --> 00:01:42.977
To help us visualize this shape here,

00:01:42.977 --> 00:01:46.528
I've kind of drawn a picture
of our coordinate plane.

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If we've viewed as an angle,

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if we're kind of above it,

00:01:49.937 --> 00:01:50.960
you can kind of start to see

00:01:50.960 --> 00:01:52.198
how this figure would look.

00:01:52.198 --> 00:01:53.960
Once again, I've drawn the base.

00:01:53.960 --> 00:01:55.911
I've drawn the base of it.

00:01:57.264 --> 00:02:00.323
I've drawn the base of
it right over there.

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Maybe I should to make it clear.

00:02:03.304 --> 00:02:04.467
Let me get like this.

00:02:04.467 --> 00:02:06.823
Shade it in kind of parallel

00:02:06.823 --> 00:02:08.398
to these cross sections.

00:02:08.398 --> 00:02:10.550
So, I've drawn the base right over there.

00:02:10.550 --> 00:02:11.990
There's two other sides.

00:02:11.990 --> 00:02:15.616
There the side that's on
... I've drawn it here.

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I guess you could view it on its top side

00:02:17.846 --> 00:02:21.076
or the left side right over there.

00:02:21.076 --> 00:02:23.194
Over on this picture that would be this.

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When we're looking at it from above.

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Then you have this other side.

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I guess on this view

00:02:29.765 --> 00:02:32.997
this one you could call
kind of the right side.

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Over here this is kind of the ...

00:02:35.028 --> 00:02:37.819
when you viewed over here
this is the bottom side.

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The whole reason why I set this up,

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and we tempting to visualize this figure.

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I want to see if you can
come up with a definite

00:02:47.683 --> 00:02:51.415
integral that describes
the volume of this figure.

00:02:51.415 --> 00:02:53.592
That kind of almost looks like a football

00:02:53.592 --> 00:02:55.357
if you cut it in half or a rugby ball.

00:02:55.357 --> 00:02:57.329
It's skewed a little bit as well.

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What's an expression a
definite integral that

00:02:59.982 --> 00:03:01.890
expresses the volume of this.

00:03:02.890 --> 00:03:04.615
I encourage you to use the fact

00:03:04.615 --> 00:03:06.063
that it intersects at the points.

00:03:06.063 --> 00:03:08.418
These functions intersect
at the point zero zero

00:03:08.418 --> 00:03:09.753
and (c,d).

00:03:09.753 --> 00:03:11.372
Can you come up with some expression

00:03:11.372 --> 00:03:13.485
a definite in row of terms of zeros,

00:03:13.485 --> 00:03:16.483
and Cs, and Ds, and Fs, and Gs

00:03:16.483 --> 00:03:18.867
that describe the volume of this figure?

00:03:20.267 --> 00:03:21.690
Assuming you've paused the video,

00:03:21.690 --> 00:03:24.113
and have had a go at it,
let's think about it.

00:03:25.036 --> 00:03:26.990
If we want to find the volume,

00:03:26.990 --> 00:03:28.426
one way to think about it is

00:03:28.610 --> 00:03:31.794
we could take the volume of,

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we could approximate the volume

00:03:33.859 --> 00:03:37.606
as the volume of these
individual triangles.

00:03:37.606 --> 00:03:40.729
That would be the area of
each of these triangles

00:03:40.729 --> 00:03:43.520
times some very small depth.

00:03:43.520 --> 00:03:45.525
Some very small depth.

00:03:46.586 --> 00:03:48.155
I'll just shade it in to show the depth.

00:03:48.155 --> 00:03:52.964
Some very small depth
which we could call DX.

00:03:52.964 --> 00:03:55.769
Once again we could find
the volume of each of these

00:03:55.769 --> 00:03:57.301
by finding the area,

00:03:57.301 --> 00:03:58.748
the cross sectional area there,

00:03:58.748 --> 00:04:00.412
and then multiplying that times

00:04:00.412 --> 00:04:02.590
a little DX.

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A little DX which would give us three,

00:04:05.218 --> 00:04:07.548
so this is a little DX,

00:04:07.548 --> 00:04:09.843
which would give us three-dimensional.

00:04:09.843 --> 00:04:13.028
That's our DX. I could write
that a little bit neater.

00:04:14.135 --> 00:04:16.499
DX to give us a little bit
of three-dimensional depth.

00:04:16.499 --> 00:04:18.292
How could we ... what is the volume of

00:04:18.292 --> 00:04:19.917
one of these figures going to look like?

00:04:20.686 --> 00:04:25.037
Well, if we say, let me
just call this height H.

00:04:25.037 --> 00:04:29.832
We know that H is going
to be F of X minus G of X.

00:04:29.832 --> 00:04:31.739
That's this distance right over here.

00:04:31.739 --> 00:04:33.252
Let's call that H.

00:04:33.252 --> 00:04:35.392
We know that H is going to be,

00:04:35.392 --> 00:04:37.579
maybe I should say H of X.

00:04:38.825 --> 00:04:40.780
It is going to be a function of X.

00:04:40.780 --> 00:04:43.485
H of X is going to be F of X.

00:04:43.485 --> 00:04:48.485
F of X minus G of X. Minus G of X.

00:04:49.028 --> 00:04:54.028
Given a H, what is going to
be the area of this triangle?

00:04:54.955 --> 00:04:57.970
This is a 45, 45, 90 triangle.

00:04:57.970 --> 00:05:00.590
This is 90, then this is going
to have to be 45 degrees.

00:05:00.590 --> 00:05:02.309
That's going to have to be 45 degrees.

00:05:02.309 --> 00:05:06.139
We know that the sides
of a 45, 45, 90 triangle

00:05:06.139 --> 00:05:08.392
or squared of two times the hypotenuse.

00:05:08.392 --> 00:05:11.666
This is going to be
squared of two over two

00:05:11.666 --> 00:05:13.267
times the hypotenuse.

00:05:13.267 --> 00:05:15.977
Squared two over two time the hypotenuse.

00:05:15.977 --> 00:05:18.659
You could get that straight
from the pythagorean theorem.

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If the side, let's say it's length A,

00:05:21.390 --> 00:05:23.463
then this side has length A.

00:05:23.463 --> 00:05:26.573
You're going to have A
squared plus A squared

00:05:26.573 --> 00:05:28.735
is equal to the hypotenuse squared,

00:05:28.735 --> 00:05:32.246
or two A squared is equal
to the hypotenuse squared.

00:05:33.384 --> 00:05:36.773
A squared is equal to the
hypotenuse squared over two,

00:05:36.773 --> 00:05:40.975
or that A is equal to H
over the squared of two,

00:05:40.975 --> 00:05:44.321
which is the same thing as
a squared of two H over two.

00:05:44.321 --> 00:05:46.046
I just rationalized the denominator,

00:05:46.046 --> 00:05:47.191
multiplied by open numerator,

00:05:47.191 --> 00:05:48.695
and then denominator by squared of two.

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That's where I got this from.

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What is the area going to be?

00:05:52.612 --> 00:05:55.125
The area is just going
to be your base times

00:05:55.125 --> 00:05:57.015
your height times one-half.

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Let me write that down.

00:05:59.442 --> 00:06:00.497
The area

00:06:01.282 --> 00:06:02.831
the area there

00:06:03.409 --> 00:06:05.347
the area is just going to be the base,

00:06:05.347 --> 00:06:09.849
which is squared of two over
two times our hypotenuse,

00:06:09.849 --> 00:06:12.438
times the height which is
squared of two over two

00:06:12.438 --> 00:06:15.453
time our hypotenuse, times one-half.

00:06:16.037 --> 00:06:17.557
Times one-half.

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If we didn't do that one-half,

00:06:18.693 --> 00:06:22.227
we'd be figuring out the
area of this entire square.

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This is obviously where
were concerned with

00:06:25.392 --> 00:06:26.595
the triangle.

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What's this going to be?

00:06:27.895 --> 00:06:29.708
This is going to be
squared of two over two

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times squared of two
over two is going to be

00:06:31.984 --> 00:06:33.925
one-half and then you're going to have

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another one-half.

00:06:35.107 --> 00:06:40.107
So, it's one-fourth, one-fourth H square.

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Did I do that right?

00:06:41.559 --> 00:06:44.141
This is ... yes it's
going to be two over four,

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which is one-half, and then
times another one-half.

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This one-fourth H squared is the area.

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Now, what's going to be the volume

00:06:51.619 --> 00:06:53.708
of each of these triangles?

00:06:54.400 --> 00:06:57.575
Well the volume of each of
these triangles right over here,

00:06:58.128 --> 00:07:02.408
the volume is just going
to be our area times DX.

00:07:02.408 --> 00:07:07.176
It's just going to be one-fourth H squared

00:07:07.176 --> 00:07:12.176
times our depth. Times our depth DX.

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If we just integrated a bunch of these,

00:07:15.219 --> 00:07:19.124
from our X equals zero
all the way to X equals C,

00:07:19.124 --> 00:07:22.452
we essentially have our
volume of the entire figure.

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So how could we write that?

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We want to write.

00:07:25.615 --> 00:07:28.875
We want to essentially to
find the volume of the figure.

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We're kind of in the home stretch here.

00:07:31.705 --> 00:07:33.114
Actually let me write volume.

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This is volume of a section.

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Volume of a cross section.

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So, what's the volume of the
entire thing going to be?

00:07:40.975 --> 00:07:42.091
The volume

00:07:43.290 --> 00:07:46.405
volume of the figure

00:07:46.405 --> 00:07:48.041
of the figure

00:07:49.164 --> 00:07:51.799
is going to be the definite integral,

00:07:52.752 --> 00:07:55.093
the definite integral from X equals zero

00:07:55.093 --> 00:08:00.093
to X equals C. X equals zero
to C of one-fourth H squared.

00:08:03.659 --> 00:08:05.356
One-fourth.

00:08:05.356 --> 00:08:08.916
We know that H is equal
to F of X minus G of X.

00:08:08.916 --> 00:08:10.313
Instead of H, I'm going to write

00:08:10.313 --> 00:08:15.313
F of X minus G of X squared

00:08:15.668 --> 00:08:18.157
squared DX.

00:08:18.963 --> 00:08:21.875
DX and we're done.

00:08:21.875 --> 00:08:24.686
We just found an expression for,

00:08:24.686 --> 00:08:26.568
a definite integral expression

00:08:26.568 --> 00:08:27.942
I guess you could say,

00:08:27.942 --> 00:08:31.119
for the volume of this strange figure

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that we have defined.


Another arc length integration example

WEBVTT
Kind: captions
Language: en

00:00:00.229 --> 00:00:01.876
- [Voiceover] This right here is the graph

00:00:01.876 --> 00:00:05.021
of Y is equal to X to the third over six

00:00:05.021 --> 00:00:07.191
plus one over two X.

00:00:07.191 --> 00:00:08.848
And what I want to do in this video is

00:00:08.848 --> 00:00:11.388
to figure out the arc
length along this curve

00:00:11.388 --> 00:00:13.660
between X equals one and X equals two.

00:00:13.660 --> 00:00:15.683
And so we've already highlighted that

00:00:15.683 --> 00:00:18.414
in this purple-ish color.

00:00:18.414 --> 00:00:20.149
So I encourage you to pause this video

00:00:20.149 --> 00:00:21.269
and try it out on your own.

00:00:21.269 --> 00:00:22.698
And I'll give you one hint.

00:00:22.698 --> 00:00:25.797
Assuming that you apply
the arc length formula

00:00:25.797 --> 00:00:29.297
correctly, it'll just be
a bit of power algebra

00:00:29.297 --> 00:00:33.388
that you'll have to do to
actually find the arc length.

00:00:33.388 --> 00:00:35.043
So I'm assuming you've had a go at it.

00:00:35.043 --> 00:00:36.691
Let's work through it together.

00:00:36.691 --> 00:00:39.937
So a few videos ago,
we got a justification

00:00:39.937 --> 00:00:42.338
for the formula of arc length.

00:00:42.338 --> 00:00:46.488
We got arc length, arc length is equal to

00:00:46.488 --> 00:00:50.211
the integral from the
lower boundary in X to

00:00:50.211 --> 00:00:52.912
the upper boundary in X,
and this is the arc length,

00:00:52.912 --> 00:00:55.157
if we're dealing in terms
of X we could actually

00:00:55.157 --> 00:00:57.081
deal in terms of other variables.

00:00:57.081 --> 00:01:00.831
And so it's going to be
one plus F prime of X

00:01:02.422 --> 00:01:03.422
squared D X.

00:01:05.672 --> 00:01:08.099
So for this particular F of
X we just need to figure out

00:01:08.099 --> 00:01:10.250
what F prime of X is, we
need to square it, we need to

00:01:10.250 --> 00:01:12.574
add one to it, and then we
have to take the square root

00:01:12.574 --> 00:01:13.439
of that.

00:01:13.439 --> 00:01:16.056
So let's do all of that step-by-step.

00:01:16.056 --> 00:01:19.056
So what is F prime of X going to be?

00:01:20.393 --> 00:01:22.524
Let's see, X to the third,
the derivative of that

00:01:22.524 --> 00:01:24.107
is three X squared.

00:01:25.058 --> 00:01:27.894
So three X squared, three
over six is going to be

00:01:27.894 --> 00:01:32.630
X squared over two, and
this is one half X to the

00:01:32.630 --> 00:01:34.800
negative one is one way to think about it.

00:01:34.800 --> 00:01:38.550
And so that's going to
be negative one half X

00:01:39.468 --> 00:01:40.789
to the negative two.

00:01:40.789 --> 00:01:44.622
Negative one half X to
the negative two power.

00:01:48.832 --> 00:01:52.436
Now, what is F prime of X squared?

00:01:52.436 --> 00:01:53.291
So it's going to be,

00:01:53.291 --> 00:01:55.816
Actually let's just write
out what is one plus F prime

00:01:55.816 --> 00:01:57.339
of X squared?

00:01:57.339 --> 00:01:59.641
So it's going to be,

00:01:59.641 --> 00:02:02.141
So one, one plus F prime of X,

00:02:05.700 --> 00:02:07.367
F prime of X squared

00:02:09.157 --> 00:02:10.532
is going to be equal to

00:02:10.532 --> 00:02:14.032
it's gonna be one plus this thing squared.

00:02:15.275 --> 00:02:17.825
And so, let's see, this
term right here squared

00:02:17.825 --> 00:02:20.765
is going to be X to the fourth over four.

00:02:20.765 --> 00:02:24.504
It's gonna be X to the fourth over four.

00:02:24.504 --> 00:02:28.439
Now the product of
these times two is going

00:02:28.439 --> 00:02:30.022
to get us negative,

00:02:32.115 --> 00:02:34.906
Let's see X squared time
X to the negative two

00:02:34.906 --> 00:02:36.929
which is going to be one.

00:02:36.929 --> 00:02:40.717
And so it's going to be negative one half.

00:02:40.717 --> 00:02:42.592
Right, I just took the
product of these two

00:02:42.592 --> 00:02:45.305
and then multiplied that by two, yep,

00:02:45.305 --> 00:02:47.356
it's just gonna be negative one half.

00:02:47.356 --> 00:02:49.190
And then this term right over here squared

00:02:49.190 --> 00:02:52.463
is going to be positive one fourth X

00:02:52.463 --> 00:02:54.783
to the negative four.

00:02:54.783 --> 00:02:57.485
One fourth X to the negative four.

00:02:57.485 --> 00:02:59.789
Now let's see we have
one and a minus one half

00:02:59.789 --> 00:03:01.214
so we can simplify them a little bit.

00:03:01.214 --> 00:03:04.685
This is going to be
equal to X to the fourth

00:03:04.685 --> 00:03:06.195
over four.

00:03:06.195 --> 00:03:10.362
One minus one half is plus
one half plus one fourth,

00:03:12.474 --> 00:03:16.224
plus one fourth X to
the negative four power.

00:03:18.059 --> 00:03:19.836
Let me make that a little bit clearer.

00:03:19.836 --> 00:03:21.669
X to the negative four power.

00:03:21.669 --> 00:03:23.775
So this seems a little bit strange.

00:03:23.775 --> 00:03:25.659
And we're going to have to
take the square root of it.

00:03:25.659 --> 00:03:28.305
But maybe we can set this
up so that it's a product

00:03:28.305 --> 00:03:30.480
of perfect squares, because it does look,

00:03:30.480 --> 00:03:33.113
you know X to fourth
and let's see if we can

00:03:33.113 --> 00:03:34.798
write this in a way
that we could recognize

00:03:34.798 --> 00:03:37.103
how to factor it a little bit better.

00:03:37.103 --> 00:03:38.480
So let's factor out a one fourth X

00:03:38.480 --> 00:03:41.284
to the negative fourth power.

00:03:41.284 --> 00:03:42.871
So this is going to be equal to

00:03:42.871 --> 00:03:46.204
one fourth X to the negative four times,

00:03:50.006 --> 00:03:53.239
let's see I factored that out,

00:03:53.239 --> 00:03:55.586
so if you factor one
fourth X to the negative

00:03:55.586 --> 00:03:57.493
fourth out of this first term, and I could

00:03:57.493 --> 00:03:58.388
color-code 'em a little bit.

00:03:58.388 --> 00:04:00.156
This first term, when you factor this out

00:04:00.156 --> 00:04:03.730
is going to be X to the eighth power.

00:04:03.730 --> 00:04:06.904
Now this term right over
here, if I factored out

00:04:06.904 --> 00:04:10.154
a one fourth, it's going to be equal to

00:04:11.546 --> 00:04:13.213
two X to the fourth.

00:04:15.130 --> 00:04:17.576
So this term right over here,

00:04:17.576 --> 00:04:20.352
once again, this powering
through the algebra part.

00:04:20.352 --> 00:04:22.685
So plus two X to the fourth,

00:04:24.080 --> 00:04:25.664
and then this one is
pretty straight forward.

00:04:25.664 --> 00:04:28.257
This is just going to be equal to one.

00:04:28.257 --> 00:04:29.257
So plus one.

00:04:31.056 --> 00:04:31.889
Plus one.

00:04:33.804 --> 00:04:36.432
And now this is looking interesting.

00:04:36.432 --> 00:04:39.332
Because this one, this right over here

00:04:39.332 --> 00:04:43.389
we could rewrite as one half
X to the negative two squared.

00:04:43.389 --> 00:04:46.227
This is equal to, that same color,

00:04:46.227 --> 00:04:49.477
one half X to the negative two squared.

00:04:51.966 --> 00:04:54.935
And this over here we
could rewrite this as X

00:04:54.935 --> 00:04:56.518
to the fourth plus,

00:04:58.046 --> 00:05:00.046
so we write it this way,

00:05:01.183 --> 00:05:04.302
this is going to be X to the fourth plus

00:05:04.302 --> 00:05:05.653
one squared.

00:05:05.653 --> 00:05:08.452
Alright, X to the fourth
times X to the fourth is

00:05:08.452 --> 00:05:09.292
X to the eighth.

00:05:09.292 --> 00:05:10.532
One squared is one.

00:05:10.532 --> 00:05:13.401
The product of them times
two is two X to the fourth.

00:05:13.401 --> 00:05:16.082
So this over here is
the same thing as that.

00:05:16.082 --> 00:05:17.902
And so now if we wanted
to take the square root

00:05:17.902 --> 00:05:19.622
of both sides.

00:05:19.622 --> 00:05:23.313
So if we wanted to say the
square root of one prime

00:05:23.313 --> 00:05:24.221
plus X squared.

00:05:24.221 --> 00:05:25.918
Notice I'm just focusing on the algebra.

00:05:25.918 --> 00:05:30.085
So what we've done so far
this is one plus F prime

00:05:31.238 --> 00:05:32.321
of X squared.

00:05:33.958 --> 00:05:37.029
Now we want to take the
square root of that.

00:05:37.029 --> 00:05:40.279
So that's going to be
the square root of this.

00:05:40.279 --> 00:05:42.809
And so that's going to be equal to

00:05:42.809 --> 00:05:45.880
square root of this is,
let me just color code it,

00:05:45.880 --> 00:05:50.047
this is going to be one half
X to the negative two power

00:05:51.413 --> 00:05:54.580
times, times X to the fourth plus one.

00:05:56.061 --> 00:05:58.644
Times X to the fourth plus one.

00:06:00.871 --> 00:06:02.954
X to the fourth plus one.

00:06:04.562 --> 00:06:05.558
I did that right, yep.

00:06:05.558 --> 00:06:06.695
Or

00:06:06.695 --> 00:06:08.818
I could actually, now that I did all this

00:06:08.818 --> 00:06:10.336
to put this in a form
that I could actually

00:06:10.336 --> 00:06:12.488
recognize, now I could
distribute things back.

00:06:12.488 --> 00:06:16.088
This is going to be equal to one half,

00:06:16.088 --> 00:06:20.161
X to the negative two
times X to the fourth

00:06:20.161 --> 00:06:21.244
is X squared.

00:06:22.312 --> 00:06:26.479
One half X squared plus one
half X to the negative two.

00:06:29.343 --> 00:06:32.433
And if we're gonna take,
if we're gonna take,

00:06:32.433 --> 00:06:34.204
Let me rewrite this.

00:06:34.204 --> 00:06:37.287
That's one plus F prime of X squared.

00:06:39.866 --> 00:06:43.228
Now let's take the definite integral.

00:06:43.228 --> 00:06:46.568
I'm going to give myself
some space to write my D X.

00:06:46.568 --> 00:06:48.232
So we're gonna take the definite integral,

00:06:48.232 --> 00:06:51.282
in this case, from X equals one

00:06:51.282 --> 00:06:53.771
X equals one to X equals two.

00:06:53.771 --> 00:06:58.224
So this is the definite
integral from X equals one

00:06:58.224 --> 00:07:00.635
to X equals two of this D X.

00:07:00.635 --> 00:07:01.877
So it's going to be the
definite integral of this

00:07:01.877 --> 00:07:05.044
from X equals one to X equals two D X.

00:07:06.478 --> 00:07:09.189
And so this is fairly straight forward.

00:07:09.189 --> 00:07:12.732
The anti-derivative of one half X squared

00:07:12.732 --> 00:07:14.352
that's going to be, what?

00:07:14.352 --> 00:07:16.994
That's going to be one,
let's see it's X to the third

00:07:16.994 --> 00:07:18.335
and then we divide by three.

00:07:18.335 --> 00:07:20.398
So one half divided by three is one sixth.

00:07:20.398 --> 00:07:22.481
One sixth X to the third.

00:07:24.233 --> 00:07:25.779
And then this is going to be,

00:07:25.779 --> 00:07:26.870
we're going to increment,

00:07:26.870 --> 00:07:28.200
this is gonna be X to the negative one.

00:07:28.200 --> 00:07:31.484
We're gonna divide by
that so minus one half X

00:07:31.484 --> 00:07:33.514
to the negative one power.

00:07:33.514 --> 00:07:34.347
Is that right?

00:07:34.347 --> 00:07:35.180
Did I do that?

00:07:35.180 --> 00:07:37.059
Yep, negative one, when you
take it, yep negative two

00:07:37.059 --> 00:07:39.500
and then this one, yep, that looks good.

00:07:39.500 --> 00:07:40.748
And we're in the homestretch.

00:07:40.748 --> 00:07:42.109
We're gonna evaluate it.

00:07:42.109 --> 00:07:43.609
At two and at one.

00:07:45.173 --> 00:07:46.022
And so we get,

00:07:46.022 --> 00:07:47.627
when you evaluate it at two, you get

00:07:47.627 --> 00:07:51.117
two to the third, which is eight over six

00:07:51.117 --> 00:07:53.617
minus one half times one half.

00:07:56.021 --> 00:07:58.362
So this is minus one fourth.

00:07:58.362 --> 00:07:59.195
And when you evaluate it at one,

00:07:59.195 --> 00:08:02.354
you're gonna have minus
so we're gonna subtract

00:08:02.354 --> 00:08:04.016
evaluating it at one,

00:08:04.016 --> 00:08:07.208
it's gonna be one sixth minus one half)

00:08:07.208 --> 00:08:09.610
And now we just have to
evaluate these fractions.

00:08:09.610 --> 00:08:12.312
So this is going to be, let's see

00:08:12.312 --> 00:08:15.234
This is, if we divide,
this is four thirds,

00:08:15.234 --> 00:08:18.901
four thirds minus one
fourth minus one sixth

00:08:20.883 --> 00:08:23.883
and then we have plus plus one half.

00:08:25.013 --> 00:08:27.887
Now let's see the common
denominator here would be 12.

00:08:27.887 --> 00:08:30.107
So this is be four thirds over 12 is

00:08:30.107 --> 00:08:31.588
16 over 12.

00:08:31.588 --> 00:08:34.381
Alright multiply the numerator
and denominator by four.

00:08:34.381 --> 00:08:38.548
Minus one fourth, the same
thing as three over 12.

00:08:40.786 --> 00:08:43.286
Now this is minus two over 12.

00:08:44.800 --> 00:08:47.613
And then we have plus six over 12.

00:08:47.613 --> 00:08:49.754
And so this is going to be equal to,

00:08:49.754 --> 00:08:51.917
I think we deserve at least
a little mini-drum roll

00:08:51.917 --> 00:08:53.318
right over here.

00:08:53.318 --> 00:08:57.485
So 16 minus three is going
to be 13 minus two is 11

00:08:59.474 --> 00:09:00.724
plus six is 17.

00:09:03.748 --> 00:09:05.090
So there we have it.

00:09:05.090 --> 00:09:06.661
The length of that arc

00:09:06.661 --> 00:09:08.383
along this curve.

00:09:08.383 --> 00:09:10.906
Between X equals one and X equals two.

00:09:10.906 --> 00:09:15.710
That length right over
there is 17, 17 twelfths.

00:09:15.710 --> 00:09:16.627
Were done.


Arc length intro

WEBVTT
Kind: captions
Language: en

00:00:00.609 --> 00:00:01.893
- [Voiceover] We've used the definite

00:00:01.893 --> 00:00:04.037
integral to find areas.

00:00:04.037 --> 00:00:05.867
What I want to do now is to see if we can

00:00:05.867 --> 00:00:09.406
use the definitive role
to find an arc length.

00:00:09.406 --> 00:00:10.934
What do I mean by that?

00:00:10.934 --> 00:00:15.291
Well, if I start at this point
on the graph of a function,

00:00:15.291 --> 00:00:17.663
and if I were to go at
this point right over here,

00:00:17.663 --> 00:00:19.625
not a straight line, we
know already how to find

00:00:19.625 --> 00:00:22.287
the distance in the straight
line but instead we want

00:00:22.287 --> 00:00:24.719
to find the distance along the curve.

00:00:24.719 --> 00:00:26.566
If we lay a string along the curve,

00:00:26.566 --> 00:00:28.562
what would be the
distance right over here?

00:00:28.562 --> 00:00:33.312
That's what I'm talking
about by arc length.

00:00:33.312 --> 00:00:36.138
What we could think about it
is okay, that's going to be

00:00:36.138 --> 00:00:41.138
from x equals a to x
equals b along this curve.

00:00:43.451 --> 00:00:45.077
So how can we do it?

00:00:45.077 --> 00:00:48.377
Well, the one thing that
integration, integral calculus is

00:00:48.377 --> 00:00:49.990
teaching us is that when we see something

00:00:49.990 --> 00:00:52.122
that's changing like
this, what we could do is

00:00:52.122 --> 00:00:55.397
we can break it up into
infinitely small parts.

00:00:55.397 --> 00:00:57.694
Infinitely small parts
that we can approximately

00:00:57.694 --> 00:01:00.275
with things like lines and
rectangles, and then we could

00:01:00.275 --> 00:01:04.081
take the infinite sum of
those infinitely small parts.

00:01:04.081 --> 00:01:06.974
So let me break up my arc length.

00:01:06.974 --> 00:01:09.194
Let me break it up into infinitely

00:01:09.194 --> 00:01:11.982
small sections of arc length.

00:01:11.982 --> 00:01:14.252
Let me call each of those
infinitely small sections

00:01:14.252 --> 00:01:18.428
of my arc length a, I
guess I could say a length

00:01:18.428 --> 00:01:23.209
to differential, an arc length
to differential, I call it ds.

00:01:23.209 --> 00:01:25.607
I'll draw it a much bigger
that when I at least

00:01:25.607 --> 00:01:27.351
I conceptualize what a differential is,

00:01:27.351 --> 00:01:28.902
just so that we could see it.

00:01:28.902 --> 00:01:31.366
What do I mean by breaking
it up into these ds's?

00:01:31.366 --> 00:01:33.059
Well, if that's the ds,

00:01:33.059 --> 00:01:34.762
and then let me do the
others in other colors,

00:01:34.762 --> 00:01:39.762
that's another infinitely
small change in my arc length.

00:01:40.554 --> 00:01:42.710
Another infinitely small
change in my arc length.

00:01:42.710 --> 00:01:46.623
If I summed all of these ds's together,

00:01:46.623 --> 00:01:48.473
I'm going to get the arc length.

00:01:48.473 --> 00:01:52.873
The arc length, if I take
is going to be the integral

00:01:52.873 --> 00:01:57.873
of all of these ds's sum
together over this integral

00:01:58.937 --> 00:02:01.782
so we can denote it like this.

00:02:01.782 --> 00:02:03.216
But this doesn't help me right now.

00:02:03.216 --> 00:02:06.798
This is in terms of this arc
length that's differential.

00:02:06.798 --> 00:02:10.127
We do know how to do things
in terms of dx's and dy's.

00:02:10.127 --> 00:02:11.305
Let's see if we can re-express this

00:02:11.305 --> 00:02:14.258
in terms of dx's and dy's.

00:02:14.258 --> 00:02:16.067
If we go on a really, really small scale,

00:02:16.067 --> 00:02:17.556
once again, we can approximate.

00:02:17.556 --> 00:02:20.066
This is going to be a line.

00:02:20.066 --> 00:02:21.425
We just the way that we approximated

00:02:21.425 --> 00:02:22.857
area with rectangles at first.

00:02:22.857 --> 00:02:24.049
But if we have an infinite number

00:02:24.049 --> 00:02:26.146
of infinite small
rectangles, we're actually

00:02:26.146 --> 00:02:29.553
approximating a non-rectangular region.

00:02:29.553 --> 00:02:31.113
The area on a non-rectangular region.

00:02:31.113 --> 00:02:32.861
Similarly, we're approximating with lines

00:02:32.861 --> 00:02:35.258
with the infinitely small
and there's infinite number

00:02:35.258 --> 00:02:38.222
of them, you are actually
finding the length of the curve.

00:02:38.222 --> 00:02:42.599
Well, just focusing on
this is a line for now.

00:02:42.599 --> 00:02:43.945
This distance right over
here, I'm just going

00:02:43.945 --> 00:02:47.019
to try express in terms of dx's and dy's.

00:02:47.019 --> 00:02:50.581
So this distance right
over here, that's dx.

00:02:50.581 --> 00:02:54.143
You can do this as
infinitely small change in x,

00:02:54.143 --> 00:02:58.189
and this distance right
over here, this is a dy.

00:02:58.189 --> 00:03:00.920
Once again, I'm being
loosey-goosey with differentials.

00:03:00.920 --> 00:03:02.218
Really giving you
conceptual understanding,

00:03:02.218 --> 00:03:04.697
not a reader's proof, but
it'll give you a sense

00:03:04.697 --> 00:03:08.156
of where the formula for arc
length is actually coming from.

00:03:08.156 --> 00:03:12.649
Based on this, you can see
the ds could be expressed

00:03:12.649 --> 00:03:15.932
based on the Pythagorean
Theorem as equal to dx squared

00:03:15.932 --> 00:03:19.271
plus dy squared, or you could rewrite this

00:03:19.271 --> 00:03:22.336
as square root of dx
squared plus dy squared.

00:03:22.336 --> 00:03:24.443
So we could rewrite this.

00:03:24.443 --> 00:03:29.051
We could say this is the same
thing as the integral of,

00:03:29.051 --> 00:03:31.365
instead of writing ds,
I'm going to write it

00:03:31.365 --> 00:03:36.365
as the square root of dx
squared plus dy squared.

00:03:45.328 --> 00:03:48.960
Once again, this is straight
out of the Pythagorean Theorem.

00:03:48.960 --> 00:03:50.304
Now this is starting to get interesting.

00:03:50.304 --> 00:03:52.140
I've written in terms of dx's and dy's

00:03:52.140 --> 00:03:53.065
but they're getting squared.

00:03:53.065 --> 00:03:54.032
They're under radical sign.

00:03:54.032 --> 00:03:56.064
What can I do to simplify this?

00:03:56.064 --> 00:04:00.358
Or at least write it in a way
that I know how to integrate.

00:04:00.358 --> 00:04:03.946
Well, I could factor out a dx squared,

00:04:03.946 --> 00:04:04.865
so let me just rewrite it.

00:04:04.865 --> 00:04:06.207
This is going to be the same thing

00:04:06.207 --> 00:04:10.952
as the integral of the square root.

00:04:10.952 --> 00:04:13.181
I'm going to factor out the dx squared,

00:04:13.181 --> 00:04:18.181
dx squared times one
plus dy over dx squared.

00:04:24.384 --> 00:04:27.256
Notice this, and this
is exact same quantity.

00:04:27.256 --> 00:04:29.597
If I distribute this dx squared,

00:04:29.597 --> 00:04:31.618
I'm going to get this right up here.

00:04:31.618 --> 00:04:34.526
Now I can take the dx
squared out of the radical.

00:04:34.526 --> 00:04:39.526
So this is going to be the integral of,

00:04:39.668 --> 00:04:42.806
let me write that in the white color,

00:04:42.806 --> 00:04:47.806
the integral of one plus dy, dx squared.

00:04:48.041 --> 00:04:50.356
This is interesting because
we know what dy, dx is.

00:04:50.356 --> 00:04:52.979
This is the derivative of our function,

00:04:52.979 --> 00:04:56.117
dy, dx squared.

00:04:56.117 --> 00:04:58.062
If you take the dx squared
out of the radical,

00:04:58.062 --> 00:05:00.917
the square root of dx squared
is just going to be dx.

00:05:00.917 --> 00:05:04.637
This is just going to be dx.

00:05:04.638 --> 00:05:06.636
This is just going to be dx.

00:05:06.636 --> 00:05:08.114
Now this is really
interesting because we know

00:05:08.114 --> 00:05:10.643
how to find this between two bounds.

00:05:10.643 --> 00:05:13.455
We can now take the
definitive role from a to b.

00:05:13.455 --> 00:05:16.641
This is now we are
integrating a bunch of dx's

00:05:16.641 --> 00:05:20.499
or we're integrating with respect to x.

00:05:20.499 --> 00:05:23.308
We could say, "Okay, x
equals a to x equals b."

00:05:23.308 --> 00:05:25.108
Let's take the sum of the product

00:05:25.108 --> 00:05:28.966
of this expression and
dx, and this is essential.

00:05:28.966 --> 00:05:31.786
This is the formula for arc length.

00:05:31.786 --> 00:05:33.116
The formula for arc length.

00:05:33.116 --> 00:05:34.601
This looks complicated.

00:05:34.601 --> 00:05:35.883
In the next video, we'll
see there's actually

00:05:35.883 --> 00:05:37.113
fairly straight forward to apply

00:05:37.113 --> 00:05:39.130
although sometimes in math gets airy.

00:05:39.130 --> 00:05:41.061
If you wanted to write this in
slightly different notation,

00:05:41.061 --> 00:05:46.061
you could write this as equal
to the integral from a to b,

00:05:46.433 --> 00:05:51.433
x equals a to x equals b of
the square root of one plus.

00:05:52.446 --> 00:05:54.622
Instead of dy, dx, I could write it

00:05:54.622 --> 00:05:59.622
as f prime of x squared, dx.

00:06:03.877 --> 00:06:06.215
So if you know the function,
if you know what f of x is,

00:06:06.215 --> 00:06:08.776
take the derivative of it
with respect to x squared

00:06:08.776 --> 00:06:11.517
added to one, take the square
root, and then multiply,

00:06:11.517 --> 00:06:14.083
and then take the definite integral

00:06:14.083 --> 00:06:18.065
of that with respect to x from a to b.

00:06:18.065 --> 00:06:20.852
We'll do that in the next video.


Worked example: arc length

WEBVTT
Kind: captions
Language: en

00:00:00.629 --> 00:00:04.317
- [Voiceover] So, right
over here, we have the graph

00:00:04.317 --> 00:00:08.828
of the function y is equal
to x to the 3/2 power.

00:00:08.828 --> 00:00:13.828
And what I wanna do is find
the arc length of this curve,

00:00:14.811 --> 00:00:18.520
from when x equals zero
to when x is equal to--

00:00:18.520 --> 00:00:20.651
and I'm gonna pick a strange number here,

00:00:20.651 --> 00:00:21.693
and I picked this strange number

00:00:21.693 --> 00:00:23.923
'cause it makes the numbers
work out very well--

00:00:23.923 --> 00:00:26.740
to x is equal to 32/9.

00:00:26.740 --> 00:00:28.428
32/9 is, let's see...

00:00:28.428 --> 00:00:33.428
That's three and 5/9, so
it's gonna be right around...

00:00:35.815 --> 00:00:37.976
So that's three and 1/2, so
it's gonna be a little bit

00:00:37.976 --> 00:00:41.801
past three and 1/2, so it's
gonna be right over there.

00:00:41.801 --> 00:00:46.801
So we wanna find this arc
length right over here,

00:00:47.222 --> 00:00:49.764
this thing that I have depicted in yellow.

00:00:49.764 --> 00:00:54.155
So it's from zero to 32/9.

00:00:54.155 --> 00:00:55.949
And I encourage you to pause the video

00:00:55.949 --> 00:00:58.788
and try this out on your own.

00:00:58.788 --> 00:01:00.055
So I'm assuming you've had a go at it.

00:01:00.055 --> 00:01:01.702
And if at any point while
I'm working through it,

00:01:01.702 --> 00:01:05.128
you feel inspired, always
feel free to pause the video

00:01:05.128 --> 00:01:07.519
and continue working with it.

00:01:07.519 --> 00:01:09.734
So let's just apply the arc length formula

00:01:09.734 --> 00:01:13.155
that we got kind of a conceptual proof for

00:01:13.155 --> 00:01:15.592
in the previous video.

00:01:15.592 --> 00:01:17.691
So we know that the arc length...

00:01:17.691 --> 00:01:18.375
Let me write this.

00:01:18.375 --> 00:01:23.375
The arc length is going to be
equal to the definite integral

00:01:25.609 --> 00:01:30.609
from zero to 32/9 of the square root...

00:01:32.185 --> 00:01:35.475
Actually, let me just write
it in general terms first,

00:01:35.475 --> 00:01:38.694
so that you can kinda see the
formula and then how we apply it.

00:01:38.694 --> 00:01:43.694
So, it's the square root of one
plus f-prime of x squared d x.

00:01:46.469 --> 00:01:50.920
And in this case, it's going
to be the definite integral

00:01:50.920 --> 00:01:55.920
from zero to 32/9 of the
square root of one plus...

00:02:00.511 --> 00:02:01.557
Now, what's the derivative?

00:02:01.557 --> 00:02:06.557
If f of x is x to the 3/2,
then f-prime of x is going

00:02:09.052 --> 00:02:13.135
to be 3/2 x to the 1/2.

00:02:13.135 --> 00:02:15.128
And we picked this particular function

00:02:15.128 --> 00:02:18.426
because it simplifies
quite well when we put it

00:02:18.426 --> 00:02:20.690
under the radical, and
it's fairly straightforward

00:02:20.690 --> 00:02:22.513
to find the anti-derivative.

00:02:22.513 --> 00:02:24.946
So we've done a lot of
engineering of this problem

00:02:24.946 --> 00:02:26.394
to make the numbers work out well,

00:02:26.394 --> 00:02:27.797
but let's just go through it.

00:02:27.797 --> 00:02:32.517
So this is f-prime of x;
f-prime of x squared is going

00:02:32.517 --> 00:02:33.984
to be this quantity squared.

00:02:33.984 --> 00:02:38.600
It's going to be 9/4 x
to the 1/2 squared is x.

00:02:38.600 --> 00:02:43.600
So, one plus 9/4 x d x.

00:02:43.716 --> 00:02:45.850
And so, now, we just
have a definite integral

00:02:45.850 --> 00:02:48.323
that we know how to
solve this type of thing.

00:02:48.323 --> 00:02:50.589
And you might be able to
even do this in your head,

00:02:50.589 --> 00:02:52.241
essentially, do the u-substitution:

00:02:52.241 --> 00:02:54.583
say I have one plus 9/4 x.

00:02:54.583 --> 00:02:56.297
Its derivative is 9/4.

00:02:56.297 --> 00:02:58.895
I can kind of engineer that if I want,

00:02:58.895 --> 00:03:02.393
but instead, I'm just going to
do straight up u-substitution.

00:03:02.393 --> 00:03:07.393
So, if I say u is equal to one plus 9/4 x,

00:03:10.729 --> 00:03:11.610
then we know...

00:03:11.610 --> 00:03:12.830
Let's see.

00:03:12.830 --> 00:03:17.830
D u d x is going to be equal to 9/4.

00:03:19.579 --> 00:03:24.579
Or, we could say d u is equal to 9/4 d x.

00:03:24.808 --> 00:03:28.094
Or, we could say d x...

00:03:28.094 --> 00:03:30.180
Let me scroll down a little bit.

00:03:30.180 --> 00:03:32.438
We could say d x is equal to...

00:03:32.438 --> 00:03:36.149
I'm just gonna multiply
both sides times 4/9.

00:03:36.149 --> 00:03:38.854
Let's go to 4/9 d u.

00:03:38.854 --> 00:03:41.487
And then we just have to change
the bounds of integration.

00:03:41.487 --> 00:03:46.487
When x is equal to zero, then
u is going to be equal to...

00:03:48.781 --> 00:03:51.262
9/4 times zero is just zero, so u is going

00:03:51.262 --> 00:03:52.936
to be equal to one.

00:03:52.936 --> 00:03:57.936
And when x is equal to 32/9--

00:03:59.975 --> 00:04:01.458
and this is why that number was picked--

00:04:01.458 --> 00:04:03.602
what's u going to be equal to?

00:04:03.602 --> 00:04:07.936
32/9 times 9/4 is gonna be 32/4,

00:04:07.936 --> 00:04:09.992
which is going to be eight plus one.

00:04:09.992 --> 00:04:13.011
So that worked out very
nicely; imagine that.

00:04:13.011 --> 00:04:13.797
So there we have it.

00:04:13.797 --> 00:04:17.636
So, this is going to be equal
to the definite integral...

00:04:17.636 --> 00:04:18.998
Actually, let me make
it clear that this is

00:04:18.998 --> 00:04:20.349
what is equal to this.

00:04:20.349 --> 00:04:25.349
The definite integral
from u is equal to one,

00:04:25.517 --> 00:04:27.589
to u is equal to 9--

00:04:27.589 --> 00:04:28.922
I'm gonna make it very
explicit that I'm dealing

00:04:28.922 --> 00:04:30.839
with u now--

00:04:30.839 --> 00:04:35.488
of the square root of u.

00:04:35.488 --> 00:04:40.152
And instead of d x, we
have d x is 4/9 d u.

00:04:40.152 --> 00:04:43.760
Let me do it this way: the square root...

00:04:43.760 --> 00:04:45.909
Whoops, that's not the right color.

00:04:45.909 --> 00:04:50.909
Square root of u; instead of
d x, we have times 4/9 d u.

00:04:52.880 --> 00:04:57.880
And I'm just gonna take the
4/9 and stick it out here.

00:04:58.178 --> 00:05:01.597
And we know how to apply the fundamental,

00:05:01.597 --> 00:05:04.329
or, I guess, the second fundamental
theorem of calculus here,

00:05:04.329 --> 00:05:06.663
to evaluate this definite integral.

00:05:06.663 --> 00:05:11.663
This is going to be 4/9
times the anti-derivative

00:05:12.446 --> 00:05:15.030
of the square root of u,
which is the same thing

00:05:15.030 --> 00:05:16.551
as u to the 1/2.

00:05:16.551 --> 00:05:19.564
It's going to be u to the
3/2, and then we divide

00:05:19.564 --> 00:05:23.204
by 3/2, which is the same
thing as multiplying by 2/3.

00:05:23.204 --> 00:05:26.371
We're going to evaluate
that at u equals nine

00:05:26.371 --> 00:05:28.886
and u is equal to one.

00:05:28.886 --> 00:05:31.427
And so, we're in the home stretch here.

00:05:31.427 --> 00:05:36.427
This is going to be equal
to 4/9 times 2/3 times 9

00:05:41.641 --> 00:05:46.641
to the 3/2 minus 2/3 times one to the 3/2.

00:05:49.636 --> 00:05:52.119
So, 9 to the 3/2, that is...

00:05:52.119 --> 00:05:54.724
Let's see: the square
root of nine is three,

00:05:54.724 --> 00:05:57.302
to the third power is 27.

00:05:57.302 --> 00:05:59.352
And this, of course, is one.

00:05:59.352 --> 00:06:03.436
So we are left with 2/3 of...

00:06:03.436 --> 00:06:04.920
Actually, let's just factor out the 2/3,

00:06:04.920 --> 00:06:07.117
that makes it easier.

00:06:07.117 --> 00:06:12.117
This is going to be equal to
2/3 times 4/9 is equal to 8/27.

00:06:15.355 --> 00:06:17.176
I've just factored out the 2/3.

00:06:17.176 --> 00:06:19.777
And then we're gonna have 27 minus one

00:06:19.777 --> 00:06:23.274
inside, I guess you could
say the brackets now.

00:06:23.274 --> 00:06:28.274
So 27 minus one is just going to be 26.

00:06:29.803 --> 00:06:32.157
Times 26.

00:06:32.157 --> 00:06:36.212
And we could obviously
simplify this more if we want.

00:06:36.212 --> 00:06:39.173
Eight times 26 is...

00:06:39.173 --> 00:06:41.154
So, actually, let's just
figure that out, just for fun.

00:06:41.154 --> 00:06:42.792
So, eight times 26 is going

00:06:42.792 --> 00:06:47.792
to be 160 plus eight times six
is 48, so it's 208 over 27.

00:06:56.910 --> 00:07:01.910
And we are done.


Examples leveraging integration properties

WEBVTT
Kind: captions
Language: en

00:00:01.070 --> 00:00:02.907
- [Voiceover] Over here, we have the graph

00:00:02.907 --> 00:00:06.082
of the function Y is equal to E to the X.

00:00:06.082 --> 00:00:08.467
Actually, let me make that clear.

00:00:08.467 --> 00:00:11.953
That right over there is
Y is equal to E to the X.

00:00:11.953 --> 00:00:13.505
This is, of course, our X axis.

00:00:13.505 --> 00:00:15.034
This is our Y axis.

00:00:15.236 --> 00:00:17.858
And so, this blue area right over here,

00:00:17.907 --> 00:00:19.766
this blue area, we can denote that

00:00:19.766 --> 00:00:22.120
as the definite integral from zero to one

00:00:22.120 --> 00:00:25.269
of E to the X, DX and let's say we know

00:00:25.269 --> 00:00:27.113
in the future, we'll learn how to come up

00:00:27.113 --> 00:00:29.282
with this number but
let's just say we know

00:00:29.282 --> 00:00:34.282
that that area is equal to E minus one.

00:00:34.762 --> 00:00:39.231
Given that, I have these
other three definite integrals

00:00:39.231 --> 00:00:40.521
and what I encourage you to do is

00:00:40.521 --> 00:00:43.065
try to pause this video
and try to evaluate

00:00:43.065 --> 00:00:45.881
what this three other
definite integrals are

00:00:45.881 --> 00:00:47.939
using only this information

00:00:47.939 --> 00:00:49.530
and what we already know

00:00:49.530 --> 00:00:52.790
about definite integral properties.

00:00:53.816 --> 00:00:58.155
I am assuming that you
have given a go at it.

00:00:58.472 --> 00:01:00.015
You might first say, okay, let's see

00:01:00.015 --> 00:01:01.440
E to the X plus two.

00:01:01.440 --> 00:01:03.461
You might be tempted to decompose this

00:01:03.461 --> 00:01:05.950
into definite integral of E to the X, DX

00:01:05.950 --> 00:01:09.097
plus the definite integral from one to--

00:01:09.097 --> 00:01:10.965
So that you could decompose
this from one to one

00:01:10.965 --> 00:01:12.916
at E to X, DX plus one to one,

00:01:12.916 --> 00:01:14.636
plus the definite integral from one to one

00:01:14.636 --> 00:01:17.793
of two DX but you have to remember

00:01:17.793 --> 00:01:18.839
if you're taking the definite integral

00:01:18.839 --> 00:01:20.089
where your bounds are the same.

00:01:20.089 --> 00:01:22.318
This is the definite
integral from one to one.

00:01:22.318 --> 00:01:24.837
This is going to be zero.

00:01:24.947 --> 00:01:27.312
This is going to be equal to zero.

00:01:27.312 --> 00:01:28.217
Once again, you can imagine

00:01:28.217 --> 00:01:29.831
that you're trying to
find the area of something

00:01:29.831 --> 00:01:31.484
that has no width.

00:01:31.484 --> 00:01:32.826
You're not moving at all.

00:01:32.826 --> 00:01:35.742
You have no width along the X axis here.

00:01:35.742 --> 00:01:38.585
That was pretty straightforward.

00:01:38.585 --> 00:01:41.744
From one to one, your
area is going to be zero.

00:01:42.114 --> 00:01:43.913
Now, let's think through
this one over here.

00:01:43.913 --> 00:01:46.108
Here my bounds are different.

00:01:46.108 --> 00:01:49.684
They are the same as these
bounds right over here.

00:01:49.684 --> 00:01:51.262
I guess that gives me a pretty good clue

00:01:51.262 --> 00:01:53.108
that maybe if I can break
this up a little bit,

00:01:53.108 --> 00:01:54.841
especially if one of these is in the form

00:01:54.841 --> 00:01:56.896
of E to the X, I might be able

00:01:56.896 --> 00:01:58.891
to evaluate this definite integral.

00:01:58.891 --> 00:02:00.061
Let's do that.

00:02:00.297 --> 00:02:01.339
The first thing I could do

00:02:01.339 --> 00:02:02.398
is I could break this up

00:02:02.398 --> 00:02:05.091
as the definite integral from zero to one

00:02:05.091 --> 00:02:08.494
of 3DX and then I could say

00:02:09.524 --> 00:02:12.522
Plus the definite
integral from zero to one

00:02:12.522 --> 00:02:15.622
of negative E to the X, DX

00:02:15.622 --> 00:02:17.377
and then I can take the negative.

00:02:17.377 --> 00:02:19.042
That negative is really
just a negative one.

00:02:19.042 --> 00:02:20.763
We saw that you could
take that scaling factor

00:02:20.763 --> 00:02:22.583
out of the integral so let's do that.

00:02:22.583 --> 00:02:25.335
This is the same thing as the
integral from zero to one.

00:02:25.335 --> 00:02:29.669
3DX minus the integral from zero to one,

00:02:29.669 --> 00:02:32.220
E to the X, DX.

00:02:32.597 --> 00:02:33.703
Well, lucky for us,

00:02:33.703 --> 00:02:36.742
we already know what, let me do this

00:02:36.742 --> 00:02:37.752
in that same color.

00:02:37.752 --> 00:02:39.157
We already know what this,

00:02:39.157 --> 00:02:40.901
I have trouble changing colors.

00:02:40.901 --> 00:02:44.706
We already know what
this right over here is.

00:02:44.706 --> 00:02:47.569
That's going to be equal to E minus one.

00:02:47.797 --> 00:02:50.591
If you subtract an E minus one,

00:02:50.591 --> 00:02:51.868
you're going to get,

00:02:51.868 --> 00:02:52.951
actually, let me just write it out

00:02:52.951 --> 00:02:54.466
so we don�t get confused.

00:02:54.466 --> 00:02:59.466
We're going to subtract an E minus one,

00:02:59.877 --> 00:03:01.478
what is this going to be equal to?

00:03:01.478 --> 00:03:04.483
The definite integral
from zero to one of 3DX.

00:03:04.684 --> 00:03:07.601
We just have to graph that to imagine it.

00:03:09.779 --> 00:03:13.563
We're going from zero to one

00:03:13.614 --> 00:03:16.429
and the function is that one, two, three.

00:03:16.429 --> 00:03:17.944
I'm not doing it at the same scale.

00:03:17.944 --> 00:03:20.225
The function looks like that.

00:03:20.735 --> 00:03:23.605
This right over here is just this area.

00:03:23.845 --> 00:03:28.845
Under the function F
of X is equal to three.

00:03:29.862 --> 00:03:30.880
from zero to one

00:03:30.880 --> 00:03:32.390
So, what's this area?

00:03:32.390 --> 00:03:33.819
Well, this is just our base.

00:03:33.819 --> 00:03:35.144
Our width is one.

00:03:35.144 --> 00:03:36.809
Our height is three.

00:03:36.809 --> 00:03:39.625
This is just going to evaluate to three.

00:03:39.625 --> 00:03:41.179
That's just going to be equal to three

00:03:41.179 --> 00:03:44.049
and so, this is going to be equal to three

00:03:44.049 --> 00:03:47.717
and you're going to have minus E,

00:03:48.058 --> 00:03:49.875
just distributing the negative side

00:03:49.875 --> 00:03:51.742
and then plus one.

00:03:54.657 --> 00:03:57.812
And then of course, three plus one is

00:03:57.812 --> 00:04:01.756
going to be equal to four.

00:04:01.756 --> 00:04:04.139
This is going to be equal to four

00:04:04.139 --> 00:04:09.139
minus four, minus E and we're done.

00:04:09.258 --> 00:04:12.251
We're able to solve or
I guess we say, evaluate

00:04:12.251 --> 00:04:14.868
or find the value of
this definite integral

00:04:14.868 --> 00:04:16.409
using only integration properties

00:04:16.409 --> 00:04:18.898
in the value of this definite integral.

00:04:18.986 --> 00:04:20.311
Now, what about this one over here?

00:04:20.311 --> 00:04:22.365
This one is interesting way.

00:04:22.396 --> 00:04:23.907
I have different bounds here.

00:04:23.907 --> 00:04:25.624
Definitely go from two to three.

00:04:25.624 --> 00:04:28.968
Having E to X minus two.

00:04:29.122 --> 00:04:30.245
And when you see something like this,

00:04:30.245 --> 00:04:31.611
especially, if you're
trying to evaluate this

00:04:31.611 --> 00:04:33.173
only with integration properties,

00:04:33.173 --> 00:04:35.944
the big thing that may or
may not jump out of you

00:04:35.944 --> 00:04:39.010
is that E to the X minus
two is shifted version

00:04:39.010 --> 00:04:39.875
of E to the X.

00:04:39.875 --> 00:04:41.637
It's shifted by two to the right.

00:04:41.637 --> 00:04:44.923
Let me draw that.

00:04:44.925 --> 00:04:46.310
Instead of this point being there,

00:04:46.310 --> 00:04:48.955
this point is going to be there

00:04:49.091 --> 00:04:53.648
and so, the curve is going
to look something like

00:04:53.648 --> 00:04:55.288
let me draw my best attempt.

00:04:55.841 --> 00:04:58.111
My best attempt to do it.

00:04:58.834 --> 00:05:01.343
It's going to be like there.

00:05:01.647 --> 00:05:02.889
It's going to be like there.

00:05:02.889 --> 00:05:06.211
It's going to look something.

00:05:07.022 --> 00:05:09.679
I'm trying my best to just shift it over.

00:05:09.765 --> 00:05:11.375
So each of these,

00:05:11.375 --> 00:05:14.741
this is going to be
shifted to right over here.

00:05:16.433 --> 00:05:17.729
There you go.

00:05:17.928 --> 00:05:20.856
E to the X minus two.

00:05:21.129 --> 00:05:23.229
We're going from two to three

00:05:23.290 --> 00:05:25.311
and so, what are the bounds now?

00:05:25.311 --> 00:05:26.530
The bounds are from two to three.

00:05:26.530 --> 00:05:29.323
We're going from this
point right over here

00:05:29.323 --> 00:05:31.889
to three, which is going to be over here.

00:05:32.640 --> 00:05:35.487
Just going to be over here.

00:05:35.489 --> 00:05:38.208
Take this point, you shift one and two.

00:05:38.208 --> 00:05:39.851
You're going to get right over here

00:05:39.851 --> 00:05:41.903
and what do we just do?

00:05:41.903 --> 00:05:44.380
We're thinking about this
area right over here.

00:05:44.380 --> 00:05:48.300
We shifted the curve
over to the right by two.

00:05:48.537 --> 00:05:51.265
This is the curve of E to the X minus two

00:05:51.465 --> 00:05:53.864
and we shifted the interval event

00:05:53.864 --> 00:05:56.295
that we're trying to find the area under.

00:05:56.295 --> 00:05:57.570
We shifted that by two.

00:05:57.570 --> 00:05:59.398
This is going to have the exact same area.

00:05:59.398 --> 00:06:01.864
I just shifted everything
to the right by two.

00:06:02.906 --> 00:06:07.088
This area is also going to be E minus one.

00:06:07.088 --> 00:06:08.573
In my mind, this one's
a little bit tricky.

00:06:08.573 --> 00:06:11.043
You just have to realize that this

00:06:11.213 --> 00:06:13.576
is just a shifted version of that

00:06:13.576 --> 00:06:16.049
and that we also shifted the two bounds

00:06:16.049 --> 00:06:17.932
to the right and we already saw that

00:06:17.932 --> 00:06:21.225
in one of the integration property videos.


Breaking up integral interval

WEBVTT
Kind: captions
Language: en

00:00:00.586 --> 00:00:02.535
- So we've depicted here the area under

00:00:02.535 --> 00:00:05.591
the curve F of X above the X-axis,

00:00:05.591 --> 00:00:09.393
between the points X
equals A and X equals B.

00:00:09.393 --> 00:00:10.997
And we've denote it as the definite

00:00:10.997 --> 00:00:14.765
integral from A to B of F of X, DX.

00:00:14.765 --> 00:00:16.474
Now what I wanna do with
this video is introduce

00:00:16.474 --> 00:00:21.042
a third value, C, that
is in between A and B.

00:00:21.042 --> 00:00:24.140
And it could be equal to A
or it could be equal to B.

00:00:24.140 --> 00:00:28.459
So let me just introduce
it, right, just like that.

00:00:28.459 --> 00:00:32.042
And I could write that A
is less than or equal to C,

00:00:32.042 --> 00:00:34.364
which is less than or equal to B.

00:00:34.364 --> 00:00:37.301
And what I wanna think
about is, how does this

00:00:37.301 --> 00:00:39.970
definite integral relate to
the definite integral from

00:00:39.970 --> 00:00:42.739
A to C and the definite
integral from C to B.

00:00:42.739 --> 00:00:44.496
So let's think through that.

00:00:44.496 --> 00:00:49.496
So we have the definite
integral from A to C of F of X.

00:00:50.944 --> 00:00:53.535
Actually I've already used
that purple color for the

00:00:53.535 --> 00:00:55.921
function itself, so we use green.

00:00:55.921 --> 00:01:00.921
So we have the integral
from A to C of F of X, DX,

00:01:02.844 --> 00:01:05.927
and that of course is
going to, that's going to

00:01:05.942 --> 00:01:09.976
represent this area right
over here, from A to C under

00:01:09.976 --> 00:01:14.463
the curve F of X, above
the X-axis, so that's that.

00:01:14.463 --> 00:01:17.865
And then we can have the integral from

00:01:17.865 --> 00:01:22.865
C to B of F of X, DX,
and that of course is

00:01:24.223 --> 00:01:28.143
going to represent this
area right over here.

00:01:28.143 --> 00:01:30.420
Well the one thing that
probably jumps out at you

00:01:30.420 --> 00:01:34.028
is that the entire area from
A to B, this entire area

00:01:34.028 --> 00:01:37.423
is just a sum of these two smaller areas.

00:01:37.423 --> 00:01:41.909
So this is just equal to
that plus that over there.

00:01:41.909 --> 00:01:43.197
And once again, you might say,

00:01:43.197 --> 00:01:45.547
"Why is this integration property useful?"

00:01:45.547 --> 00:01:49.455
That if I found a C that is in
this interval that's greater

00:01:49.455 --> 00:01:52.882
than or equal to A and it's
less than or equal to B, "Why

00:01:52.882 --> 00:01:56.723
"is it useful to be able to
break up the integral this way?"

00:01:56.723 --> 00:01:58.679
Well as you'll see this is
really useful, it can be

00:01:58.679 --> 00:02:01.086
very useful when you're
doing disc, when you're

00:02:01.086 --> 00:02:03.684
looking at functions that
have discontinuities,

00:02:03.684 --> 00:02:05.451
if they have step
functions you can break up

00:02:05.451 --> 00:02:07.789
the larger integral
into smaller integrals.

00:02:07.789 --> 00:02:10.097
You'll also see that this
is useful when we prove

00:02:10.097 --> 00:02:12.152
the fundamental theorem of Calculus.

00:02:12.152 --> 00:02:13.148
So in general, this is actually a

00:02:13.148 --> 00:02:14.854
very, very, very useful technique.

00:02:14.854 --> 00:02:17.983
Let me actually draw an
integral where it might be

00:02:17.983 --> 00:02:22.299
very useful to utilize that property.

00:02:22.299 --> 00:02:27.299
So if this is A, this is B,
and let's say the function,

00:02:28.932 --> 00:02:31.348
I'm just gonna make it
constant over an interval.

00:02:31.348 --> 00:02:34.209
So it's constant from there
to there, and then it's,

00:02:34.209 --> 00:02:36.287
and then it drops down
from there to there.

00:02:36.287 --> 00:02:38.792
Let's say the function looked like this.

00:02:38.792 --> 00:02:42.143
Well you could say that the
larger integral, which would be

00:02:42.143 --> 00:02:44.692
the area under the curve,
it would be all of this.

00:02:44.692 --> 00:02:45.876
Let's just say it's a gap right

00:02:45.876 --> 00:02:48.269
there or it jumps down there.

00:02:48.269 --> 00:02:52.165
So this entire area you
can break up into two,

00:02:52.165 --> 00:02:55.580
you can break up into two smaller areas.

00:02:55.580 --> 00:02:57.836
So you could break that up into that area

00:02:58.098 --> 00:03:01.340
right over there, and then this area right

00:03:01.340 --> 00:03:04.433
over here using this integral property.


Definite integral of shifted function

WEBVTT
Kind: captions
Language: en

00:00:00.915 --> 00:00:03.730
- Let's say that we know that
this area under the curve,

00:00:03.730 --> 00:00:08.166
y is equal to f of x, let me
label it, y is equal to f of x.

00:00:08.566 --> 00:00:11.798
So under this curve, above
the x axis, between a and b,

00:00:11.798 --> 00:00:14.098
which we denote is the
definite integral from a to b,

00:00:14.098 --> 00:00:15.501
of f of x, d of x.

00:00:15.516 --> 00:00:16.932
Let's say we know what this is.

00:00:16.932 --> 00:00:21.087
Let's say it's equal to, let's
say this area is equal to five.

00:00:21.934 --> 00:00:26.150
So given that, can you
figure out what the definite

00:00:26.150 --> 00:00:30.136
integral, what the definite integral from

00:00:32.475 --> 00:00:35.174
a plus some constant, c.

00:00:35.174 --> 00:00:37.065
Let me do this in a different color.

00:00:37.481 --> 00:00:40.466
So, what is going to be
the definite integral of

00:00:40.466 --> 00:00:45.466
f of x minus c dx, from
a plus c, to b plus c?

00:00:50.466 --> 00:00:52.312
So, this might look a little daunting.

00:00:52.497 --> 00:00:54.216
But I encourage you to kind of try to

00:00:54.216 --> 00:00:55.683
visualize what's going to happen here.

00:00:55.683 --> 00:00:57.878
Try to pick a c in your
brain and try to graph them.

00:00:58.124 --> 00:00:59.647
And pause the video and try to think about

00:00:59.647 --> 00:01:02.857
what this is going to be,
given what we know about this.

00:01:03.965 --> 00:01:05.913
So I'm assuming you've had a go at it.

00:01:06.283 --> 00:01:08.756
So what is f of x minus c?

00:01:09.218 --> 00:01:11.966
Well that's essentially
the function f of x,

00:01:11.966 --> 00:01:14.270
shifted to the right by c.

00:01:14.747 --> 00:01:16.278
So, let's do that.

00:01:16.648 --> 00:01:18.146
So that's going to look like,

00:01:18.300 --> 00:01:20.467
so if we take that function,
we shift it to the right.

00:01:20.683 --> 00:01:24.618
Let's say that this distance
right over here is c.

00:01:24.865 --> 00:01:26.633
So if you shift it to the right by c,

00:01:26.633 --> 00:01:29.574
it's going to look something like that.

00:01:29.898 --> 00:01:31.963
So I just copied and
pasted by original one.

00:01:31.963 --> 00:01:33.518
It's going to look something like that.

00:01:33.518 --> 00:01:35.892
And I can even color code it.

00:01:36.216 --> 00:01:38.682
So this thing right over here,

00:01:38.682 --> 00:01:42.816
this is the graph of, this is the graph of

00:01:42.816 --> 00:01:47.816
y is equal to, y is
equal to f of x minus c.

00:01:49.482 --> 00:01:51.050
And so all I did,

00:01:51.050 --> 00:01:54.688
it really just shifts
everything over by c.

00:01:54.688 --> 00:01:59.688
It just shifted everything over by c.

00:02:07.119 --> 00:02:08.207
And this is something you probably learned

00:02:08.207 --> 00:02:10.631
in precalculus class or in algebra class.

00:02:10.816 --> 00:02:12.448
And the key thing to realize is, okay,

00:02:12.448 --> 00:02:15.914
when x is equal to c, you're
essentially inputting.

00:02:15.914 --> 00:02:20.914
So when x is equal to c,
you're inputting zero,

00:02:21.851 --> 00:02:23.153
'cause you're gonna get c minus c.

00:02:23.153 --> 00:02:25.882
Your gonna input zero into f.

00:02:25.898 --> 00:02:30.008
So you're gonna get the
same value here, when f.

00:02:30.008 --> 00:02:32.665
So when x is equal to c for x minus c,

00:02:32.665 --> 00:02:34.265
you're inputting zero into the function.

00:02:34.265 --> 00:02:35.523
You're going to get the same value there

00:02:35.523 --> 00:02:36.837
as when you just took the function

00:02:36.837 --> 00:02:38.483
and you just inputted zero into it.

00:02:38.483 --> 00:02:40.592
So that's some of the
logic why when you take

00:02:40.592 --> 00:02:44.866
x minus c, you're shifting
to the right by c.

00:02:44.882 --> 00:02:46.443
Now let's think about the bounds here.

00:02:46.443 --> 00:02:49.850
This is a plus c, so a plus, let's shift,

00:02:49.850 --> 00:02:54.850
so a plus c is gonna get us
right over there, roughly.

00:02:55.882 --> 00:03:00.882
And so, this is a plus c and
b plus c is gonna get us,

00:03:01.342 --> 00:03:04.663
is gonna get us right over here.

00:03:04.663 --> 00:03:07.550
So this point right over here is b plus c.

00:03:07.550 --> 00:03:11.513
So our new bounds, our
new bounds look like this.

00:03:11.513 --> 00:03:16.513
Our new bounds, we're gonna
go from a plus c, to b plus c.

00:03:16.734 --> 00:03:20.183
And so this is the area, actually
let me do it that yellow color,

00:03:20.183 --> 00:03:23.811
since that's what I made
the definite integral in.

00:03:23.812 --> 00:03:26.390
So we care about this area now.

00:03:26.390 --> 00:03:28.226
We care about this area.

00:03:28.226 --> 00:03:31.217
And I think it might, it's
starting to jump out at you

00:03:31.217 --> 00:03:32.871
what this is going to be equal to.

00:03:32.886 --> 00:03:36.159
This right over here is
going to be this exact thing.

00:03:36.159 --> 00:03:37.510
We just shifted everything.

00:03:37.510 --> 00:03:38.957
We shifted the function to the right.

00:03:38.957 --> 00:03:40.703
We shifted the bounds to the right.

00:03:40.703 --> 00:03:42.444
And so this is going to be the same thing

00:03:42.444 --> 00:03:46.821
as the integral from a to b of f of x dx,

00:03:46.821 --> 00:03:49.216
which in this case, and I
just kinda made that up,

00:03:49.216 --> 00:03:51.344
in this case is going to be equal to five.

00:03:51.344 --> 00:03:52.564
But the important thing to realize,

00:03:52.564 --> 00:03:54.467
and this is a tricky one,
you'll see this sometimes,

00:03:54.467 --> 00:03:56.333
sometimes in math competitions,

00:03:56.333 --> 00:03:57.612
or in kind of a difficult test.

00:03:57.612 --> 00:03:59.735
But sometimes it can actually
help you solve an integral.

00:03:59.735 --> 00:04:02.583
Once again, if you're tackling a really,

00:04:02.583 --> 00:04:03.767
as we'll see in the future,

00:04:03.767 --> 00:04:05.576
we'll tackle some really
interesting problems,

00:04:05.576 --> 00:04:09.576
where identifying this can be valuable.

00:04:09.576 --> 00:04:11.533
You might say, "Hey, wait,
this is some bizarre thing.

00:04:11.533 --> 00:04:12.812
"How do I figure this out?"

00:04:12.812 --> 00:04:15.165
And you just realize this
is just shifting this,

00:04:15.165 --> 00:04:19.265
this is just shifting this
area over to the right by c.

00:04:19.265 --> 00:04:24.172
So it's going to have
the exact same value.


Integrating sums of functions

WEBVTT
Kind: captions
Language: en

00:00:00.797 --> 00:00:02.481
- So we have a couple of functions here.

00:00:02.481 --> 00:00:05.681
This is the graph of y is equal to f of x.

00:00:05.681 --> 00:00:08.563
This is the graph of y is equal to g of x.

00:00:08.563 --> 00:00:10.180
And we already know something.

00:00:10.180 --> 00:00:12.879
Or we know ways to represent the area

00:00:12.879 --> 00:00:15.112
under the curve y equals f of x.

00:00:15.112 --> 00:00:18.487
Between these two points x is equal to a,

00:00:18.487 --> 00:00:19.514
and x is equal to b.

00:00:19.514 --> 00:00:24.514
So this area, right over here,

00:00:24.963 --> 00:00:28.597
between the curve and the x axis,

00:00:28.597 --> 00:00:30.598
between x equals a and x equals b,

00:00:30.598 --> 00:00:33.648
we know we can write that
as the definite integral,

00:00:33.648 --> 00:00:38.648
from a to b, of f of x, dx.

00:00:38.880 --> 00:00:40.931
And we can do the same thing over here.

00:00:40.931 --> 00:00:44.146
We could call this area,

00:00:44.146 --> 00:00:46.179
let me pick a color that I have not used.

00:00:46.179 --> 00:00:47.848
Well, this is slightly different green.

00:00:47.848 --> 00:00:49.680
I could call this area right over here,

00:00:49.680 --> 00:00:53.346
the area right under the
curve y is equal to g of x.

00:00:53.346 --> 00:00:55.430
And above the positive x axis,

00:00:55.430 --> 00:00:57.713
between x equals a and x equals b,

00:00:57.713 --> 00:00:59.931
we could call that the definite integral.

00:00:59.931 --> 00:01:04.931
From a to b of g of x, dx.

00:01:05.063 --> 00:01:08.047
Now given these two things,

00:01:08.047 --> 00:01:10.429
let's actually think about
the area under the curve

00:01:10.429 --> 00:01:13.181
of the function created by the
sum of these two functions.

00:01:13.181 --> 00:01:14.663
So what do I mean by that?

00:01:14.663 --> 00:01:18.014
So let me, this is
actually a fun thing to do.

00:01:18.014 --> 00:01:19.313
Let me start again.

00:01:19.313 --> 00:01:21.298
That's exactly what we have over here.

00:01:21.298 --> 00:01:24.313
This is the graph of y is equal to f of x.

00:01:24.313 --> 00:01:25.963
But what I want to do,
is I want to approximate

00:01:25.963 --> 00:01:30.764
the graph of y is equal
to, so my goal is to graph

00:01:30.764 --> 00:01:35.764
y is equal to f of x, plus g of x.

00:01:43.395 --> 00:01:46.046
So for any given x,
it's going to be f of x.

00:01:46.046 --> 00:01:47.015
So that's that the f of x.

00:01:47.015 --> 00:01:48.578
And then I'm gonna add the g of x to it.

00:01:48.578 --> 00:01:50.097
So what will that look like?

00:01:50.097 --> 00:01:52.431
So that's gonna look like, let's see.

00:01:52.431 --> 00:01:54.979
When x is zero, g of x looks like

00:01:54.979 --> 00:01:56.114
it's about that length right there.

00:01:56.114 --> 00:01:57.630
I'm obviously approximating it.

00:01:57.630 --> 00:01:59.797
So I'm gonna have to add
that length right over here.

00:01:59.797 --> 00:02:01.646
So it'll probably be right around there.

00:02:01.646 --> 00:02:04.730
At x equals a, it's a little bit more.

00:02:04.730 --> 00:02:08.913
And so but now my f of x curve
is gone, or has increased.

00:02:08.913 --> 00:02:10.714
But I take that same distance above it.

00:02:10.714 --> 00:02:12.714
If I add the g of x there,

00:02:12.714 --> 00:02:14.598
it gets me right about there.

00:02:14.598 --> 00:02:16.613
Once again I'm just eyeballing it,

00:02:16.613 --> 00:02:17.848
trying to get an approximation,

00:02:17.848 --> 00:02:21.146
giving me an intuition actually
for f of x, plus g of x is.

00:02:21.146 --> 00:02:24.014
I'm just trying to add g of x for given x.

00:02:24.014 --> 00:02:25.946
Now let's see if I'm a little bit,

00:02:25.946 --> 00:02:27.964
let's say that I'm between a and b,

00:02:27.964 --> 00:02:32.230
g of x is about that
distance right over there.

00:02:32.230 --> 00:02:37.230
So, if I wanted to put that
same distance right over here,

00:02:37.945 --> 00:02:42.945
it gets me right about there.

00:02:44.230 --> 00:02:48.148
And then when x is equal to
b, g of x is about that big,

00:02:48.148 --> 00:02:51.996
so I have to add that
length, which is about,

00:02:51.996 --> 00:02:54.279
looks something like that.

00:02:54.279 --> 00:02:55.729
That actually looks a little bit too much.

00:02:55.729 --> 00:02:58.130
Maybe something like that.

00:02:58.130 --> 00:03:02.680
So, if I were to add
the two, I'd get a curve

00:03:02.680 --> 00:03:06.980
that looks something like this.

00:03:06.980 --> 00:03:08.847
And maybe it just keeps on
going higher and higher.

00:03:08.847 --> 00:03:10.862
So, this is the curve.

00:03:10.862 --> 00:03:12.630
Or it's a pretty good
approximation of the curve

00:03:12.630 --> 00:03:15.541
of f of x plus g of x.

00:03:15.541 --> 00:03:20.452
Now an interesting question
is, is what would be

00:03:20.612 --> 00:03:23.265
well, you know how we
can represent this area.

00:03:23.265 --> 00:03:26.930
So the area under the
curve f of x plus g of x,

00:03:26.930 --> 00:03:29.830
above the positive x axis,

00:03:29.830 --> 00:03:34.180
between x equals a and x equals b,

00:03:34.180 --> 00:03:37.064
we know we can represent that as,

00:03:37.064 --> 00:03:40.978
let me see, I have not used pink, yet.

00:03:40.978 --> 00:03:43.464
So, this area right over here,

00:03:43.464 --> 00:03:46.313
we know that that could be represented

00:03:46.313 --> 00:03:47.502
as the definite integral,

00:03:47.502 --> 00:03:52.502
from a to b of f of x, plus g of x, dx.

00:03:57.680 --> 00:03:59.015
Now the question is,

00:03:59.015 --> 00:04:01.297
how does this thing relate to d,

00:04:01.297 --> 00:04:02.714
or how does this area relate

00:04:02.714 --> 00:04:04.713
to these areas right over here?

00:04:04.713 --> 00:04:06.513
Well the important thing to realize is

00:04:06.513 --> 00:04:08.513
this area that we have in yellow,

00:04:08.513 --> 00:04:13.131
that's going to be this
area, right over here.

00:04:13.131 --> 00:04:16.130
That one's pretty clear.

00:04:16.130 --> 00:04:20.647
But how does this area in green
relate to this area there?

00:04:20.647 --> 00:04:22.063
And to think about that,

00:04:22.063 --> 00:04:25.164
we just have to think about
well what does an integral mean?

00:04:25.164 --> 00:04:27.096
What does it represent?

00:04:27.096 --> 00:04:28.446
We've already thought about these like

00:04:28.446 --> 00:04:30.864
these really small rectangles.

00:04:30.864 --> 00:04:34.180
And we're taking the
sum of the limit of an

00:04:34.180 --> 00:04:37.413
infinite number of these
infinitely thin rectangles.

00:04:37.413 --> 00:04:39.099
But when we're thinking
about Riemann sums,

00:04:39.099 --> 00:04:41.648
we're thinking about okay
we have some change in x,

00:04:41.648 --> 00:04:44.829
and then you multiply it
times essentially the height,

00:04:44.829 --> 00:04:45.687
which is going to be the value

00:04:45.687 --> 00:04:47.896
of the function at that point.

00:04:47.896 --> 00:04:51.698
Well over here, you could
have the same change in x.

00:04:51.698 --> 00:04:53.713
You can have the exact same change in x.

00:04:53.713 --> 00:04:56.198
And what is the height right over here?

00:04:56.198 --> 00:04:58.299
Well, that's going to be this
exact height right over here.

00:04:58.299 --> 00:05:00.015
You saw that when we constructed it.

00:05:00.015 --> 00:05:03.298
This is going to be the
g of x at that value.

00:05:03.298 --> 00:05:05.680
So even though the
rectangles look like they're

00:05:05.680 --> 00:05:06.981
kinda shifted around a little bit,

00:05:06.981 --> 00:05:09.563
and they're actually all
shifted up by the f of x,

00:05:09.563 --> 00:05:11.180
the heights of these rectangles

00:05:11.180 --> 00:05:12.881
that I'm drawing right over here,

00:05:12.881 --> 00:05:14.930
are exactly the same thing as the heights

00:05:14.930 --> 00:05:18.347
of the rectangles that
I'm drawing over here.

00:05:18.347 --> 00:05:22.097
They, once again, they're
all just shifted up and down

00:05:22.097 --> 00:05:24.715
by this f of x function.

00:05:24.715 --> 00:05:27.247
But these are the exact same rectangles.

00:05:27.247 --> 00:05:28.848
And they have the exact same heights.

00:05:28.848 --> 00:05:31.265
And the limit as you get
more and more of these

00:05:31.265 --> 00:05:32.264
by making them thinner and thinner,

00:05:32.264 --> 00:05:33.480
is gonna be the same
as the limit as you get

00:05:33.480 --> 00:05:35.630
more and more of these as
you get thinner and thinner.

00:05:35.630 --> 00:05:37.129
And so this area right over here,

00:05:37.129 --> 00:05:38.797
and I'm obviously not
doing a rigorous proof,

00:05:38.797 --> 00:05:40.298
I'm giving you the intuition for it,

00:05:40.298 --> 00:05:44.847
is the exact same thing as
this area right over here.

00:05:44.847 --> 00:05:46.447
So the area under this curve,

00:05:46.447 --> 00:05:50.598
the definite integral from a
to b of f of x plus g of x, dx,

00:05:50.598 --> 00:05:55.598
is just going to be the sum of
these two definite integrals.

00:05:57.848 --> 00:05:59.901
And you might say, "Oh this is obvious".

00:05:59.901 --> 00:06:01.297
Or maybe it's not so obvious.

00:06:01.297 --> 00:06:03.364
But when is this actually useful?

00:06:03.364 --> 00:06:05.241
Well, as you later actually learn

00:06:05.241 --> 00:06:06.397
to evaluate these integrals,

00:06:06.397 --> 00:06:07.629
you'll see that one of
the most powerful ideas

00:06:07.629 --> 00:06:09.563
is being able to decompose
them in this way.

00:06:09.563 --> 00:06:14.297
To say, okay, if I'm taking
the definite integral from

00:06:14.297 --> 00:06:19.297
zero to one of x squared, plus sine of x,

00:06:19.814 --> 00:06:21.563
which you may or may not
have learned to do so far,

00:06:21.563 --> 00:06:23.362
you can at least start to break this down.

00:06:23.362 --> 00:06:25.030
You can say okay, well this
is going to be the same

00:06:25.030 --> 00:06:29.796
thing as the integral from
zero to one of x squared, dx,

00:06:29.796 --> 00:06:34.796
plus the integral from zero
to one of sine of x, dx.

00:06:35.847 --> 00:06:37.180
And you see this is one
of the most powerful

00:06:37.180 --> 00:06:40.837
principles of definite
integrals, to make them,

00:06:40.837 --> 00:06:42.162
when you start to try to compute them,

00:06:42.162 --> 00:06:45.536
or even sometimes conceptualize
what they're representing.


Switching bounds of definite integral

WEBVTT
Kind: captions
Language: en

00:00:00.477 --> 00:00:02.192
- [Voiceover] We've
already seen one definition

00:00:02.192 --> 00:00:04.268
of the definite integral, and many of them

00:00:04.268 --> 00:00:06.082
are closely related to this definition

00:00:06.082 --> 00:00:09.129
that we've already seen
is the definite integral

00:00:09.129 --> 00:00:11.968
from a to b of f of x d
of x is this area shaded

00:00:11.968 --> 00:00:14.893
in blue, and we can approximate it

00:00:14.893 --> 00:00:19.893
by splitting it into n rectangles.

00:00:20.610 --> 00:00:24.139
So let's say that's the
first rectangle, one.

00:00:24.139 --> 00:00:26.428
That's the second
rectangle, two, and you're

00:00:26.428 --> 00:00:28.773
going to go all the way
to the nth rectangle,

00:00:28.773 --> 00:00:31.406
so this would be the n
minus oneth rectangle.

00:00:31.406 --> 00:00:33.147
For the sake of this
argument I'm going to make

00:00:33.147 --> 00:00:34.310
in this video, we're going to assume

00:00:34.310 --> 00:00:35.717
that they're all the same width.

00:00:35.717 --> 00:00:38.088
So this is the nth rectangle.

00:00:38.088 --> 00:00:40.634
They all have the same
width, and we see they're

00:00:40.634 --> 00:00:42.081
definitions of integration
where you don't have

00:00:42.081 --> 00:00:44.066
to have the same width here, but let's say

00:00:44.066 --> 00:00:46.371
that each of those widths are delta x,

00:00:46.371 --> 00:00:49.608
and the way that we
calculate delta x is we take

00:00:49.608 --> 00:00:54.608
b minus a and we divide it by n,

00:00:56.948 --> 00:00:58.560
which is common sense, or this is

00:00:58.560 --> 00:00:59.781
what you learned in division.

00:00:59.781 --> 00:01:01.224
We're just taking this
length and dividing it

00:01:01.224 --> 00:01:06.224
by n to get n equals spacings of delta x.

00:01:07.738 --> 00:01:08.865
So if you do this, you'll say, okay,

00:01:08.865 --> 00:01:10.739
well we see this multiple times,

00:01:10.739 --> 00:01:12.723
you can approximate it.

00:01:12.723 --> 00:01:15.773
You can approximate this
area using these rectangles

00:01:15.773 --> 00:01:19.936
as the sum from i equals one to n.

00:01:19.936 --> 00:01:24.831
So you're summing n of
these rectangle's areas

00:01:24.831 --> 00:01:28.614
where the height of
each of these rectangles

00:01:28.614 --> 00:01:33.388
are going to be f of
x sub i, where x sub i

00:01:33.388 --> 00:01:35.971
is the point at which
you're taking the function

00:01:35.971 --> 00:01:37.748
value to find out its height.

00:01:37.748 --> 00:01:41.041
So that could be x of
one, x of two, x of three,

00:01:41.041 --> 00:01:43.793
so on and so forth, and
you're multiplying that

00:01:43.793 --> 00:01:46.917
times your delta x.

00:01:46.917 --> 00:01:49.727
So you take x sub two, f of x sub two

00:01:49.727 --> 00:01:54.727
is that height right there.

00:01:55.455 --> 00:01:58.734
You multiply it times delta x.

00:01:58.734 --> 00:01:59.745
You get the area.

00:01:59.745 --> 00:02:02.347
We saw that when we looked at Riemann sums

00:02:02.347 --> 00:02:03.456
and using that to approximate.

00:02:03.456 --> 00:02:07.728
We said, hey the one definition
of the definite integral

00:02:07.728 --> 00:02:09.658
is that since this is
the area this is going

00:02:09.658 --> 00:02:14.653
to be the limit as n approaches infinity

00:02:14.653 --> 00:02:19.653
of this where delta x is defined as that.

00:02:19.747 --> 00:02:22.035
So let me just copy and paste that.

00:02:25.286 --> 00:02:27.757
So that's one way to think about it.

00:02:27.757 --> 00:02:32.757
Now, given this definition,
what do you think

00:02:33.412 --> 00:02:35.428
this, or maybe another
way to think about it,

00:02:35.428 --> 00:02:38.175
how do you think this
expression that I'm writing

00:02:38.175 --> 00:02:41.714
right over here based on this definition

00:02:41.714 --> 00:02:44.207
should relate to this expression?

00:02:44.207 --> 00:02:47.967
So notice, all I've done
is I've segued from a to b.

00:02:47.967 --> 00:02:52.086
I'm now going from b to a.

00:02:52.086 --> 00:02:54.127
How do you think these
two things should relate?

00:02:54.127 --> 00:02:56.023
I encourage you to look at all of this

00:02:56.023 --> 00:02:57.103
to come to that conclusion,

00:02:57.103 --> 00:02:59.772
and pause the video to do so.

00:02:59.772 --> 00:03:01.744
Well let's just think about
what's going to happen.

00:03:01.744 --> 00:03:05.395
This is going to be,
if I were to literally

00:03:05.395 --> 00:03:10.395
just take this and copy
and paste it, which is

00:03:10.610 --> 00:03:14.057
exactly what I'm going to
do, if I just took this,

00:03:14.057 --> 00:03:17.308
by definition, since I
swapped these two bounds,

00:03:17.308 --> 00:03:19.903
I'm going to want to swap these two.

00:03:19.903 --> 00:03:23.028
Instead of b minus a it's
going to be a minus b now.

00:03:23.028 --> 00:03:28.028
It's going to be a minus b.

00:03:30.555 --> 00:03:32.549
This value right over here.

00:03:32.549 --> 00:03:34.644
Let me make these color-coded maybe.

00:03:34.644 --> 00:03:39.644
So this orange delta x is
going to be the negative

00:03:40.992 --> 00:03:45.313
of this green delta x.

00:03:45.313 --> 00:03:48.584
This is the negative of
that right over there,

00:03:48.584 --> 00:03:50.936
and everything else is the same.

00:03:50.936 --> 00:03:52.817
So what am I going to end up doing?

00:03:52.817 --> 00:03:54.244
Well I'm essentially
going to end up having

00:03:54.244 --> 00:03:56.569
the negative value of this.

00:03:56.569 --> 00:04:00.080
So this is going to be
equal to the negative

00:04:00.080 --> 00:04:05.080
of the integral from a to b of f of x dx.

00:04:05.842 --> 00:04:08.312
So this is the result
we get, which is another

00:04:08.312 --> 00:04:10.463
really important integration property,

00:04:10.463 --> 00:04:13.908
that if you swap the
bounds of integration,

00:04:13.908 --> 00:04:15.604
and it really just comes from this idea,

00:04:15.604 --> 00:04:18.070
instead of delta x being
b minus a, if you swap

00:04:18.070 --> 00:04:20.316
the bounds of integration,
it's going to be a minus b.

00:04:20.316 --> 00:04:22.079
We're going to get the negative delta x,

00:04:22.079 --> 00:04:23.753
or the negative of your original delta x,

00:04:23.753 --> 00:04:24.907
which is going to give you the negative

00:04:24.907 --> 00:04:27.555
of this original value right over here.

00:04:27.555 --> 00:04:29.836
Once again, this is a
really, really useful

00:04:29.836 --> 00:04:31.317
integration property where you're trying

00:04:31.317 --> 00:04:32.905
to make sense of some integrals and even

00:04:32.905 --> 00:04:35.310
sometimes solve some of them.


Integrating scaled version of function

WEBVTT
Kind: captions
Language: en

00:00:00.675 --> 00:00:02.061
- [Voiceover] We've already
seen and you're probably

00:00:02.061 --> 00:00:04.527
getting tired of me
pointing it out repeatedly,

00:00:04.527 --> 00:00:07.445
that this yellow area right
over here, this area under

00:00:07.445 --> 00:00:09.550
the curve y is equal
to f of x and above the

00:00:09.550 --> 00:00:11.737
positive x-axis or I
guess I can say just above

00:00:11.737 --> 00:00:16.737
the x-axis between x
equals a and x equals b,

00:00:18.302 --> 00:00:20.470
that we can denote this area
right over here as the definite

00:00:20.470 --> 00:00:23.568
integral of from a to b of f of x dx.

00:00:24.122 --> 00:00:27.184
Now what I want to explore
in this video and it'll

00:00:27.184 --> 00:00:30.084
come up with kind of an
answer that you probably

00:00:30.084 --> 00:00:31.550
could have guessed on
your own, but at least get

00:00:31.550 --> 00:00:33.695
an intuition for it,
is that I want to start

00:00:33.695 --> 00:00:35.950
thinking about the area under the curve

00:00:35.950 --> 00:00:37.961
that's a scaled version of f of x.

00:00:38.141 --> 00:00:40.902
Let's say it's y is
equal to c times f of x.

00:00:41.354 --> 00:00:45.266
Y is equal to some number times

00:00:45.266 --> 00:00:47.116
f of x, so it's scaling f of x.

00:00:47.446 --> 00:00:49.836
And so I want this to be
kind of some arbitrary

00:00:49.836 --> 00:00:52.038
number, but just to help me
visualize, you know I have

00:00:52.038 --> 00:00:53.882
to draw something so I'm
just gonna kind of in my

00:00:53.882 --> 00:00:55.739
head let's just pretend the c is

00:00:55.739 --> 00:00:57.653
a three for visualization purposes.

00:00:58.055 --> 00:01:00.190
So it's going to be three
times, so instead of one,

00:01:00.190 --> 00:01:01.488
instead of this far right over here

00:01:01.488 --> 00:01:02.964
it's going to be about this far.

00:01:03.130 --> 00:01:05.381
For right over here, instead
of this far right over

00:01:05.381 --> 00:01:09.114
here it's going to be that
and another right over there.

00:01:09.502 --> 00:01:13.495
And then instead of it's
going to be about there.

00:01:13.702 --> 00:01:16.917
And then instead of it
being like that it's going

00:01:16.917 --> 00:01:21.078
to be one, two and then
three, right around there.

00:01:21.537 --> 00:01:23.690
So I'm starting to get a
sense of what this curve

00:01:23.690 --> 00:01:28.568
is going to look like, a
scaled version of f of x.

00:01:28.697 --> 00:01:30.966
And at least what I'm
drawing is pretty close to

00:01:30.966 --> 00:01:34.250
three times f of x, but
just to give you an idea

00:01:34.250 --> 00:01:35.959
is going to look something like,

00:01:35.959 --> 00:01:37.741
and let's see over here if this distance,

00:01:37.741 --> 00:01:39.486
do a second one, a third
one, is gonna be up here.

00:01:39.493 --> 00:01:42.511
It's gonna look something like this.

00:01:43.111 --> 00:01:44.915
It's gonna look something like that.

00:01:45.282 --> 00:01:47.916
So this is a scaled version
and the scale I did right here

00:01:47.916 --> 00:01:52.916
I assumed a positive c greater than zero,

00:01:53.660 --> 00:01:55.679
but this is just for
visualization purposes.

00:01:56.326 --> 00:01:58.811
Now what do we think the area under this

00:01:58.811 --> 00:02:01.570
curve is going to be between a and b?

00:02:01.877 --> 00:02:06.877
So what do we think this area

00:02:10.126 --> 00:02:11.766
right over here is going to be?

00:02:12.432 --> 00:02:14.390
Now we already know how we can denote it.

00:02:14.598 --> 00:02:17.602
That area right over there
is equal to the definite

00:02:17.602 --> 00:02:21.139
integral from a to b of the function

00:02:21.139 --> 00:02:25.312
we're integrating is c f of x dx.

00:02:26.035 --> 00:02:29.045
I guess to make the question
a little bit clearer,

00:02:29.045 --> 00:02:31.278
how does this relate to this?

00:02:31.352 --> 00:02:34.921
How does this green area
relate to this yellow area?

00:02:35.836 --> 00:02:38.025
Well one way to think
about it is we just scaled

00:02:38.025 --> 00:02:42.294
the vertical dimension up by
c, so one way that you could

00:02:42.294 --> 00:02:45.574
reason it is if I'm finding
the area of something,

00:02:46.190 --> 00:02:49.803
if I have the area of a
rectangle and I have the

00:02:49.803 --> 00:02:54.573
vertical dimension is
let's say I don't want to

00:02:54.573 --> 00:02:56.476
use those same letters
over and over again.

00:02:56.520 --> 00:02:59.359
Well let's say the
vertical dimension is alpha

00:02:59.359 --> 00:03:01.889
and the horizontal dimension is beta.

00:03:01.889 --> 00:03:06.643
We know that the area is
going to be alpha times beta.

00:03:07.047 --> 00:03:10.776
Now if I scale up the
vertical dimension by c,

00:03:10.776 --> 00:03:15.776
so instead of alpha this
is c times alpha and

00:03:16.339 --> 00:03:20.538
this is, the width is beta,
if I scale up the vertical

00:03:20.538 --> 00:03:25.520
dimension by c so this
is now c times alpha,

00:03:25.520 --> 00:03:26.640
what's the area going to be?

00:03:26.956 --> 00:03:31.094
Well it's going to be c alpha times beta,

00:03:31.094 --> 00:03:32.591
or another way to think of it,

00:03:34.374 --> 00:03:36.767
when I scale one of the
dimensions by c I take my

00:03:36.767 --> 00:03:40.127
old area and I scale
up my old area up by c.

00:03:40.423 --> 00:03:41.449
And that's what we're doing, we're scaling

00:03:41.449 --> 00:03:43.241
up the vertical dimension by c.

00:03:43.241 --> 00:03:44.847
When you multiply c times f of x,

00:03:44.847 --> 00:03:46.723
f of x is giving us the vertical height.

00:03:46.745 --> 00:03:48.937
Now obviously that
changes as our x changes,

00:03:48.937 --> 00:03:50.829
but when you think back
to the Reimann sums

00:03:50.829 --> 00:03:53.469
the f of x was what gave us
the height of our rectangles.

00:03:53.765 --> 00:03:56.730
We're now scaling up the
height or scaling I should say

00:03:56.730 --> 00:03:59.190
because we might be scaling
down depending on the c.

00:03:59.393 --> 00:04:02.408
We're scaling it, we're
scaling one dimension by c.

00:04:02.408 --> 00:04:03.982
If you scale one dimension by c

00:04:03.982 --> 00:04:05.681
you're gonna scale the area by c.

00:04:06.086 --> 00:04:08.973
So this right over here, the integral,

00:04:08.973 --> 00:04:10.081
let me just rewrite it.

00:04:10.321 --> 00:04:15.255
The integral from a to b of c f of x dx,

00:04:15.255 --> 00:04:17.407
that's just going to be
the scaled, we're just

00:04:17.407 --> 00:04:20.707
going to take the area of f of x,

00:04:20.707 --> 00:04:23.071
so let me do that in the same color.

00:04:23.963 --> 00:04:25.320
We're going to take the
area under the curve

00:04:25.320 --> 00:04:29.462
f of x from a to b f of x dx and

00:04:29.462 --> 00:04:34.462
we're just going to scale it up by this c.

00:04:37.207 --> 00:04:39.164
So you might say, "Okay
maybe I could have felt

00:04:39.164 --> 00:04:41.571
"that was, you know, if I
have a c inside the integral

00:04:41.571 --> 00:04:43.206
"now I can take the c
out of the integral",

00:04:43.206 --> 00:04:46.513
and once again this is
not a rigorous proof based

00:04:46.513 --> 00:04:49.044
on the definition of
the definite integral,

00:04:49.044 --> 00:04:50.184
but it hopefully gives you a little

00:04:50.184 --> 00:04:52.589
bit of intuition why you can do this.

00:04:52.905 --> 00:04:55.414
If you scale up the
function, you're essentially

00:04:55.414 --> 00:04:57.900
scaling up the vertical
dimension, so the area

00:04:57.900 --> 00:05:00.939
under this is going to just be a scaled up

00:05:00.939 --> 00:05:05.054
version of the area under
the original function f of x.

00:05:05.274 --> 00:05:07.560
And once again really,
really, really useful

00:05:07.560 --> 00:05:12.281
property of definite
integrals that's going to

00:05:12.281 --> 00:05:14.497
help us solve a bunch
of definite integrals.

00:05:14.631 --> 00:05:18.933
And kind of clarify what
we're even doing with them.


Definite integral over a single point

WEBVTT
Kind: captions
Language: en

00:00:00.633 --> 00:00:02.232
- [Voiceover] We've already
taken definite integrals

00:00:02.232 --> 00:00:05.354
and we've seen how they
represent or denote

00:00:05.354 --> 00:00:08.825
the area under a function
between two points

00:00:08.825 --> 00:00:11.143
and above the X axis.

00:00:11.143 --> 00:00:13.065
But let's do something interesting.

00:00:13.065 --> 00:00:18.065
Let's think about a definite
integral of F of X DX,

00:00:18.136 --> 00:00:21.032
It's the area under the curve,
F of X, but instead of it

00:00:21.032 --> 00:00:23.768
being mean between two different X values,

00:00:23.768 --> 00:00:26.330
say A and B like we see in multiple times,

00:00:26.330 --> 00:00:28.185
let's say it's between the same one.

00:00:28.185 --> 00:00:30.841
Let's say it's between C and C.

00:00:30.841 --> 00:00:32.938
Let's say C is right over here.

00:00:32.938 --> 00:00:35.241
What do you think this
thing right over here

00:00:35.241 --> 00:00:36.922
is going to be equal to?

00:00:36.922 --> 00:00:37.977
What does this represent?

00:00:37.977 --> 00:00:39.096
What is this equal to?

00:00:39.096 --> 00:00:39.841
I encourage you to pause the video

00:00:39.841 --> 00:00:42.370
and try to think about it.

00:00:42.370 --> 00:00:44.836
Well if you try to visualize
it, you're thinking,

00:00:44.836 --> 00:00:48.339
well the area under the curve F of X,

00:00:48.339 --> 00:00:53.339
above the X axis, from X
equals C to X equals C.

00:00:53.619 --> 00:00:55.972
So this region, I guess we could call it,

00:00:55.972 --> 00:00:57.874
that we think about
it, does have a height.

00:00:57.874 --> 00:01:00.962
The height here is F of C.

00:01:00.962 --> 00:01:03.363
What's the width?

00:01:03.363 --> 00:01:06.611
Well there is no width,
we're just at a single point.

00:01:06.611 --> 00:01:10.419
We're not going from C
to C plus some delta X

00:01:10.419 --> 00:01:14.881
or C plus some even very small change in X

00:01:14.881 --> 00:01:18.752
or C plus some other very small a value.

00:01:18.752 --> 00:01:20.688
We're just, saying at the point C.

00:01:20.688 --> 00:01:25.688
When we're thinking
about area we're thinking

00:01:26.371 --> 00:01:29.356
about how much two-dimensional
space you're taking up.

00:01:29.356 --> 00:01:33.364
But this idea, this is
just a one-dimensional,

00:01:33.364 --> 00:01:35.453
I think you could think
of it as a line segment.

00:01:35.453 --> 00:01:37.694
What's the area of a line segment?

00:01:37.694 --> 00:01:39.900
Well a line segment has no area.

00:01:39.900 --> 00:01:41.503
So this thing right over here

00:01:41.565 --> 00:01:45.122
is going to be equal to zero.

00:01:45.122 --> 00:01:47.360
Now you might say, I get that.

00:01:47.360 --> 00:01:48.705
I see why that could make sense,

00:01:48.705 --> 00:01:50.680
why that makes intuitive sense.

00:01:50.680 --> 00:01:53.720
I'm trying to find the area of a rectangle

00:01:53.720 --> 00:01:56.825
where I know it's height,
but it's width is zero.

00:01:56.825 --> 00:01:58.777
So that areas going to be zero

00:01:58.777 --> 00:02:00.585
is one way to think about it.

00:02:00.585 --> 00:02:04.281
But Sal, why are you even
pointing this out to me?

00:02:04.281 --> 00:02:06.537
As well see, especially
when we do more complex

00:02:06.537 --> 00:02:10.264
definite integration
problems and solving things

00:02:10.510 --> 00:02:15.113
sometimes recognizing this will help you

00:02:15.113 --> 00:02:20.113
simplify an integration
problem dramatically.

00:02:20.456 --> 00:02:22.888
Or you could work to be able
to get to a point like this

00:02:22.888 --> 00:02:24.392
so that you can cancel things out.

00:02:24.392 --> 00:02:26.409
Or you can say, hey that
thing right over there

00:02:26.535 --> 00:02:30.349
is just going to be equal to zero.