Force vs. time graphs

WEBVTT
Kind: captions
Language: en

00:00:00.391 --> 00:00:01.671
- [Voiceover] There's
a miniature rocketship

00:00:01.671 --> 00:00:04.195
and it's full of tiny
aliens that just got done

00:00:04.195 --> 00:00:07.268
investigating a new moon with lunar pools

00:00:07.268 --> 00:00:10.314
and all kinds of organic new life forms.

00:00:10.314 --> 00:00:12.800
But they're done investigating,
so they're gonna blast off

00:00:12.800 --> 00:00:15.732
and take their findings home
to tell all their friends.

00:00:15.732 --> 00:00:18.039
Let's say at some moment
during their ascent,

00:00:18.039 --> 00:00:20.227
they're moving at four meters per second.

00:00:20.227 --> 00:00:21.207
And they're tiny aliens,

00:00:21.207 --> 00:00:25.034
their spaceship is only 2.9 kilograms,

00:00:25.034 --> 00:00:26.826
but they need to know,
are they gonna be able

00:00:26.826 --> 00:00:28.375
to get off this moon or not?

00:00:28.375 --> 00:00:29.957
So they gotta pay
attention to their speed,

00:00:29.957 --> 00:00:32.142
but instead of using a speedometer,

00:00:32.142 --> 00:00:35.478
they clever aliens, they use
a force versus time graph.

00:00:35.478 --> 00:00:37.952
So on their dashboard, they've
got a force versus time graph

00:00:37.952 --> 00:00:40.958
and it tells them what
the net force is on them,

00:00:40.958 --> 00:00:44.107
so let's say this is the net
force, not just any force,

00:00:44.107 --> 00:00:47.104
but this is the total force
on them from rocket boosters

00:00:47.104 --> 00:00:49.913
and the force of gravity
and whatever other forces

00:00:49.913 --> 00:00:52.672
there might be, they've
got advanced force sensors.

00:00:52.672 --> 00:00:54.646
I mean come on, they can
determine their net force,

00:00:54.646 --> 00:00:56.339
let's say, and it gives them this force

00:00:56.339 --> 00:00:57.799
as a function of time.

00:00:57.799 --> 00:01:00.581
But they want to know what
is their velocity gonna be

00:01:00.581 --> 00:01:02.372
after nine seconds?

00:01:02.372 --> 00:01:04.319
So they check their force
versus time readout,

00:01:04.319 --> 00:01:05.615
and this is the graph they get.

00:01:05.615 --> 00:01:06.883
And now they can determine it.

00:01:06.883 --> 00:01:07.785
And here's how they do it.

00:01:07.785 --> 00:01:09.614
They say, "All right
there's a force, a net force

00:01:09.614 --> 00:01:13.631
"of three newtons acting
for the first four seconds."

00:01:13.631 --> 00:01:15.890
So during this entire first four seconds

00:01:15.890 --> 00:01:18.610
there's a constant force of three newtons.

00:01:18.610 --> 00:01:20.312
And every alien worth his weight knows

00:01:20.312 --> 00:01:24.479
that force, the net force
multiplied by the time duration

00:01:25.633 --> 00:01:27.584
during which that force is applied,

00:01:27.584 --> 00:01:29.340
gives you the net impulse.

00:01:29.340 --> 00:01:30.544
So this gives us the net impulse.

00:01:30.544 --> 00:01:34.032
If we take this constant
three newtons that acts,

00:01:34.032 --> 00:01:37.171
multiplied by four seconds
during which it acts,

00:01:37.171 --> 00:01:40.053
we get that there's an
impulse of 12 newton seconds.

00:01:40.053 --> 00:01:42.543
And you might be like, "Wait, who cares

00:01:42.543 --> 00:01:45.402
"about newton seconds
here, I want the velocity.

00:01:45.402 --> 00:01:46.811
"I don't want the force and the time,

00:01:46.811 --> 00:01:50.040
"I want to know the
velocity at nine seconds."

00:01:50.040 --> 00:01:52.794
But they teach you at
the alien space academy,

00:01:52.794 --> 00:01:55.067
that the net impulse is not only equal

00:01:55.067 --> 00:01:57.302
to the net force times the time,

00:01:57.302 --> 00:02:00.863
it's also equal to the change
in momentum of the object

00:02:00.863 --> 00:02:02.525
that the force was exerted on.

00:02:02.525 --> 00:02:03.422
And this is good.

00:02:03.422 --> 00:02:04.978
We know the mass of the object.

00:02:04.978 --> 00:02:06.733
We wanna know something about velocity.

00:02:06.733 --> 00:02:08.880
So we know momentum is M times V,

00:02:08.880 --> 00:02:11.541
this net impulse is
gonna help us get there.

00:02:11.541 --> 00:02:13.288
But his 12 newton seconds was only

00:02:13.288 --> 00:02:14.832
for the first four seconds.

00:02:14.832 --> 00:02:17.199
How do we figure it out
for the next three seconds?

00:02:17.199 --> 00:02:18.032
Look at this.

00:02:18.032 --> 00:02:20.781
During the next three seconds,
there's not a constant force,

00:02:20.781 --> 00:02:23.640
this force is varying, the
force is getting smaller.

00:02:23.640 --> 00:02:24.486
So how do I do this?

00:02:24.486 --> 00:02:26.080
The force isn't a constant value,

00:02:26.080 --> 00:02:29.019
so I can't just simply
take force times time,

00:02:29.019 --> 00:02:31.617
because, I mean, what force do I pick?

00:02:31.617 --> 00:02:33.356
So we're gonna use a trick.

00:02:33.356 --> 00:02:35.370
We're gonna use a trick,
because if you notice

00:02:35.370 --> 00:02:38.028
for this first section,
for the first four seconds,

00:02:38.028 --> 00:02:40.441
we took the force and multiplied

00:02:40.441 --> 00:02:42.909
by the time interval, four seconds.

00:02:42.909 --> 00:02:44.771
So what we did really is we just took

00:02:44.771 --> 00:02:47.235
the height of this rectangle times

00:02:47.235 --> 00:02:48.791
the width of this rectangle

00:02:48.791 --> 00:02:50.871
and that gives us the
area of this rectangle.

00:02:50.871 --> 00:02:52.933
So what we really did is we found the area

00:02:52.933 --> 00:02:55.313
under the force versus time graph.

00:02:55.313 --> 00:02:58.648
That gave us the impulse and
that's not a coincidence.

00:02:58.648 --> 00:03:02.815
The impulse equals the area
under a force versus time graph,

00:03:04.058 --> 00:03:05.666
and this is extremely useful to know,

00:03:05.666 --> 00:03:08.949
because now in this section,
where the force was varying,

00:03:08.949 --> 00:03:11.529
we can still use this, we
can just find the impulse

00:03:11.529 --> 00:03:14.916
by determining the area under that curve.

00:03:14.916 --> 00:03:18.130
And by area under the curve,
we mean from the line curve,

00:03:18.130 --> 00:03:21.486
in general, to the x-axis,
which, in this case,

00:03:21.486 --> 00:03:24.846
the x-axis is of the time axis.

00:03:24.846 --> 00:03:25.679
So let's do this.

00:03:25.679 --> 00:03:27.925
We found the impulse
for this first section.

00:03:27.925 --> 00:03:29.919
That was 12 newton seconds.

00:03:29.919 --> 00:03:32.620
Now we can find the impulse
for this next section

00:03:32.620 --> 00:03:34.378
by just determining the area.

00:03:34.378 --> 00:03:35.376
So this is a triangle.

00:03:35.376 --> 00:03:40.221
We'll do 1/2 base, the base
is one, two, three seconds,

00:03:40.221 --> 00:03:43.058
and the height is still three newtons.

00:03:43.058 --> 00:03:46.891
So we get a net impulse
of 4.5 newton seconds.

00:03:48.044 --> 00:03:49.572
So we've got one more section to go,

00:03:49.572 --> 00:03:51.262
but this one's a little
weird, this one's located,

00:03:51.262 --> 00:03:54.744
the area is located below the time axis,

00:03:54.744 --> 00:03:56.333
so this is still a triangle,

00:03:56.333 --> 00:03:58.350
but since the forces are negative,

00:03:58.350 --> 00:04:01.082
this is gonna count as
a negative net impulse.

00:04:01.082 --> 00:04:03.062
So when the area lies above the time axis,

00:04:03.062 --> 00:04:04.797
it counts as a positive impulse,

00:04:04.797 --> 00:04:06.966
and when the area lies
below the time axis,

00:04:06.966 --> 00:04:08.858
it counts as a negative net impulse.

00:04:08.858 --> 00:04:10.362
So how much negative net impulse?

00:04:10.362 --> 00:04:11.787
We still find the area.

00:04:11.787 --> 00:04:15.456
So the area of a triangle,
again, is gonna be 1/2,

00:04:15.456 --> 00:04:17.682
the base this time is two seconds,

00:04:17.682 --> 00:04:20.422
and the height is negative two.

00:04:20.422 --> 00:04:23.780
So negative two newtons,
which gives us a net impulse

00:04:23.780 --> 00:04:26.116
of negative two newton seconds.

00:04:26.116 --> 00:04:27.585
And now we can figure out the velocity

00:04:27.585 --> 00:04:30.733
of the spaceship at nine seconds.

00:04:30.733 --> 00:04:34.381
So, assuming that this force
readout started at this moment

00:04:34.381 --> 00:04:38.018
right over here at t equals zero seconds

00:04:38.018 --> 00:04:40.665
was the moment when it was
going four meters per second,

00:04:40.665 --> 00:04:44.441
then we could say the total
net impulse should equal

00:04:44.441 --> 00:04:47.374
the total change in
momentum of this spaceship.

00:04:47.374 --> 00:04:49.328
And we can find the total net impulse

00:04:49.328 --> 00:04:52.428
by just adding up all
the individual impulses.

00:04:52.428 --> 00:04:53.988
So during the first four seconds,

00:04:53.988 --> 00:04:56.405
there was 12 newton seconds of impulse.

00:04:56.405 --> 00:04:58.044
During the next three seconds

00:04:58.044 --> 00:05:00.972
there was 4.5 newton seconds of impulse.

00:05:00.972 --> 00:05:02.257
And during this last portion

00:05:02.257 --> 00:05:05.461
there was negative two
newton seconds of impulse,

00:05:05.461 --> 00:05:06.854
which if you add all of those up,

00:05:06.854 --> 00:05:09.697
12 plus 4.5 plus negative
two, you're gonna get

00:05:09.697 --> 00:05:13.030
positive 14.5 newton seconds of impulse.

00:05:14.502 --> 00:05:16.899
That's good news for our
alien buddies over here,

00:05:16.899 --> 00:05:18.552
they need to get off this moon,

00:05:18.552 --> 00:05:21.426
which means they need positive
impulse, upward impulse.

00:05:21.426 --> 00:05:22.648
They got some positive impulse.

00:05:22.648 --> 00:05:24.284
Let's see what their final velocity was.

00:05:24.284 --> 00:05:27.033
We know that delta P is
the change in momentum,

00:05:27.033 --> 00:05:31.145
so this is final momentum
minus initial momentum,

00:05:31.145 --> 00:05:32.249
which we could write as

00:05:32.249 --> 00:05:36.774
mass times v final minus
mass times v initial.

00:05:36.774 --> 00:05:38.834
Now, if this were an earth
rocket, this would be hard,

00:05:38.834 --> 00:05:41.234
'cause earth rockets
using earth technology,

00:05:41.234 --> 00:05:44.735
eject fuel at a huge rate out the backend,

00:05:44.735 --> 00:05:46.171
and that loses mass.

00:05:46.171 --> 00:05:48.451
That means this mass
isn't gonna stay constant.

00:05:48.451 --> 00:05:51.526
So earth rockets
essentially push fuel down

00:05:51.526 --> 00:05:53.192
which causes and equal and opposite force

00:05:53.192 --> 00:05:55.112
back on the rocket upward.

00:05:55.112 --> 00:05:57.833
But if you're losing mass, this
mass doesn't stay constant,

00:05:57.833 --> 00:05:59.624
and this whole process is a lot harder,

00:05:59.624 --> 00:06:02.940
because m final and m initial
aren't gonna be the same.

00:06:02.940 --> 00:06:05.251
Maybe this is where the phrase,
"It's not rocket science"

00:06:05.251 --> 00:06:07.534
comes from, 'cause rocket
science is a little harder

00:06:07.534 --> 00:06:08.833
when that mass changes.

00:06:08.833 --> 00:06:12.080
So let's just say, these
clever aliens can eject

00:06:12.080 --> 00:06:14.928
only a little bit of fuel, they do so.

00:06:14.928 --> 00:06:16.054
You might say, "How?"

00:06:16.054 --> 00:06:18.319
Well, there's gotta be a certain
amount of momentum, right,

00:06:18.319 --> 00:06:21.083
that they eject to give
themselves momentum up.

00:06:21.083 --> 00:06:23.823
But let's say they can
eject only a small mass

00:06:23.823 --> 00:06:26.577
at a huge speed, so there's not much fuel

00:06:26.577 --> 00:06:27.786
that they're losing.

00:06:27.786 --> 00:06:30.829
The fuel that they eject is
ejected at enormous speed,

00:06:30.829 --> 00:06:32.842
so that they get their momentum upward,

00:06:32.842 --> 00:06:35.405
but they lose almost no mass.

00:06:35.405 --> 00:06:36.869
And that let's us solve this problem

00:06:36.869 --> 00:06:39.221
assuming that the mass is constant.

00:06:39.221 --> 00:06:41.390
So if we do that, if we
assume the mass is constant,

00:06:41.390 --> 00:06:45.458
we get positive 14.5
newton seconds equals,

00:06:45.458 --> 00:06:47.562
we can pull the mass out,
the mass is a constant,

00:06:47.562 --> 00:06:49.141
so we could just write it as

00:06:49.141 --> 00:06:51.808
m times v final minus v initial,

00:06:53.369 --> 00:06:55.626
since I can pull out a common factor of m,

00:06:55.626 --> 00:06:58.649
which means I can write
this as 2.9 kilograms

00:06:58.649 --> 00:07:01.759
multiplied by the final
velocity after nine seconds

00:07:01.759 --> 00:07:03.692
and I know it's after nine
seconds, 'cause I added up

00:07:03.692 --> 00:07:06.136
all the impulse during the nine seconds,

00:07:06.136 --> 00:07:09.688
minus the initial velocity
was four meters per second.

00:07:09.688 --> 00:07:13.004
So if I divide both
sides by 2.9 kilograms,

00:07:13.004 --> 00:07:17.389
14.5 over 2.9 is 5 and
that will be newton seconds

00:07:17.389 --> 00:07:21.144
over kilograms which has
units of meters per second,

00:07:21.144 --> 00:07:22.155
and it's positive.

00:07:22.155 --> 00:07:24.633
And that's gonna equal
the leftover over here

00:07:24.633 --> 00:07:28.622
is gonna be v final minus
four meters per second.

00:07:28.622 --> 00:07:31.506
And now finally if I add
four meters per second

00:07:31.506 --> 00:07:35.623
to both sides, I get the
v final somewhere up here,

00:07:35.623 --> 00:07:40.319
v final of this rocket is gonna
be nine meters per second.

00:07:40.319 --> 00:07:41.692
So after nine seconds

00:07:41.692 --> 00:07:44.123
it ended up going nine meters per second.

00:07:44.123 --> 00:07:46.268
That's just a numerical coincidence.

00:07:46.268 --> 00:07:49.103
The way you find it, so
recapping the way we did this,

00:07:49.103 --> 00:07:51.335
we found the area under the curve,

00:07:51.335 --> 00:07:52.696
'cause the area under a curve

00:07:52.696 --> 00:07:55.160
under our force versus
time graph represents

00:07:55.160 --> 00:07:57.276
the impulse on the object.

00:07:57.276 --> 00:07:58.664
We found it for the entire trip,

00:07:58.664 --> 00:08:00.642
noting that underneath the time axis,

00:08:00.642 --> 00:08:02.825
when this curve goes under the time axis,

00:08:02.825 --> 00:08:04.821
the net impulse is gonna be negative.

00:08:04.821 --> 00:08:06.275
We added up all the net impulse,

00:08:06.275 --> 00:08:08.861
we set it equal to the change in momentum,

00:08:08.861 --> 00:08:10.608
we plugged in our values and we solved

00:08:10.608 --> 00:08:13.467
for our velocity after nine seconds.


Definite integral of absolute value function

WEBVTT
Kind: captions
Language: en

00:00:00.387 --> 00:00:02.219
- [Voiceover] So we
have f of x being equal

00:00:02.219 --> 00:00:04.984
to the absolute value of x plus two.

00:00:04.984 --> 00:00:07.345
And we wanna evaluate
the definite integral

00:00:07.345 --> 00:00:09.903
from negative four to zero of f of x, dx.

00:00:09.903 --> 00:00:11.651
And like always, pause this video

00:00:11.651 --> 00:00:13.364
and see if you can work through this.

00:00:13.364 --> 00:00:14.867
Now when you first do this you might

00:00:14.867 --> 00:00:16.133
stumble around a little bit, because

00:00:16.133 --> 00:00:17.677
how do you take the anti-derivative

00:00:17.677 --> 00:00:19.550
of an absolute value function?

00:00:19.550 --> 00:00:21.949
And the key here is to,
one way to approach it

00:00:21.949 --> 00:00:25.244
is to rewrite f of x
without the absolute value

00:00:25.244 --> 00:00:26.705
and we can do that by rewriting it

00:00:26.705 --> 00:00:28.705
as a piecewise function.

00:00:29.718 --> 00:00:31.908
And the way I'm gonna
do it, I'm gonna think

00:00:31.908 --> 00:00:35.833
about intervals where whatever we take

00:00:35.833 --> 00:00:37.989
inside the absolute value's
going to be positive

00:00:37.989 --> 00:00:39.743
and other intervals where everything

00:00:39.743 --> 00:00:41.205
that we take inside the absolute value

00:00:41.205 --> 00:00:43.077
is going to be negative.

00:00:43.077 --> 00:00:45.723
And the point at which we change

00:00:45.723 --> 00:00:48.978
is where x plus two is equal to zero

00:00:48.978 --> 00:00:51.212
or x is equal to negative two.

00:00:51.212 --> 00:00:52.592
So let's just think about the intervals

00:00:52.592 --> 00:00:56.620
x is less than negative
two and x is greater than

00:00:56.620 --> 00:00:59.181
or equal to negative two.

00:00:59.181 --> 00:01:00.403
And this could have been less than

00:01:00.403 --> 00:01:02.189
or equal, in which case this
would have been greater than,

00:01:02.189 --> 00:01:03.976
either way it would
have been equal to this

00:01:03.976 --> 00:01:06.863
absolute vale, this is a
continuous function here.

00:01:06.863 --> 00:01:09.427
And so when, let's do the easier case.

00:01:09.427 --> 00:01:11.944
When x is greater than
or equal to negative two

00:01:11.944 --> 00:01:15.240
then x plus two is going to be positive,

00:01:15.240 --> 00:01:16.863
or it's going to be greater than

00:01:16.863 --> 00:01:19.666
or equal to zero, and so
the absolute value of it

00:01:19.666 --> 00:01:21.781
is just going to be x plus two.

00:01:21.781 --> 00:01:23.569
So it's going to be x plus two

00:01:23.569 --> 00:01:26.011
when x is greater than
or equal to negative two.

00:01:26.011 --> 00:01:28.859
And what about when x is
less than negative two?

00:01:28.859 --> 00:01:31.340
Well when x is less than negative two,

00:01:31.340 --> 00:01:34.672
x plus two is going to be negative,

00:01:34.672 --> 00:01:36.257
and then if you take the absolute value

00:01:36.257 --> 00:01:38.044
of a negative number you're gonna take

00:01:38.044 --> 00:01:39.423
the opposite of it.

00:01:39.423 --> 00:01:43.330
So this is going to be
negative x plus two.

00:01:43.330 --> 00:01:46.173
And to really help grok
this, 'cause frankly

00:01:46.173 --> 00:01:48.247
this is the hardest part
of what we're doing,

00:01:48.247 --> 00:01:50.077
and really this is more
algebra than calculus.

00:01:50.077 --> 00:01:52.510
Let me draw the absolute value function

00:01:52.510 --> 00:01:54.179
to make this clear.

00:01:54.179 --> 00:01:57.429
So that is my x-axis, that is my y-axis

00:02:00.326 --> 00:02:03.826
and let's say we're here at negative two.

00:02:03.826 --> 00:02:07.781
And so when we are less
than the negative two,

00:02:07.781 --> 00:02:10.790
when x is less than negative two my graph

00:02:10.790 --> 00:02:12.376
is going to look like this.

00:02:12.376 --> 00:02:14.876
It is going to look something,

00:02:18.784 --> 00:02:20.951
it's gonna look like that.

00:02:21.952 --> 00:02:25.169
And when we are greater than negative two,

00:02:25.169 --> 00:02:27.569
do that in a different
color, when we are greater

00:02:27.569 --> 00:02:31.486
than negative two it's
going to look like this.

00:02:32.652 --> 00:02:34.442
It's going to look like that.

00:02:34.442 --> 00:02:37.497
And so notice this is in blue we have,

00:02:37.497 --> 00:02:39.775
this is the graph x plus two, we can say

00:02:39.775 --> 00:02:41.850
this is a graph of y equals x plus two.

00:02:41.850 --> 00:02:44.127
And what we have in
magenta right over here,

00:02:44.127 --> 00:02:47.504
this is the graph of negative x minus two.

00:02:47.504 --> 00:02:49.538
It has a negative slope and we intercept

00:02:49.538 --> 00:02:51.788
the y-axis at negative two.

00:02:52.662 --> 00:02:53.598
So it makes sense.

00:02:53.598 --> 00:02:55.881
There's multiple ways that
you could reason through this.

00:02:55.881 --> 00:02:57.426
Now once we break it up then we can

00:02:57.426 --> 00:02:58.852
break up the integral.

00:02:58.852 --> 00:03:00.355
We could say that what we wrote here,

00:03:00.355 --> 00:03:02.060
this is equal to the integral

00:03:02.060 --> 00:03:06.160
from negative four to
two, sorry negative four

00:03:06.160 --> 00:03:10.160
to negative two of f of
x, which is in that case

00:03:11.324 --> 00:03:14.577
it's going to be negative x minus two,

00:03:14.577 --> 00:03:17.592
I just distributed the
negative sign there.

00:03:17.592 --> 00:03:20.842
Dx, and then plus the definite integral

00:03:24.143 --> 00:03:27.357
going from negative two to zero

00:03:27.357 --> 00:03:29.107
of x plus two, dx.

00:03:32.159 --> 00:03:33.335
And just to make sure we know

00:03:33.335 --> 00:03:36.184
what we're doin' here,
if this is negative four

00:03:36.184 --> 00:03:39.441
right over here, this is
zero, that first integral

00:03:39.441 --> 00:03:43.024
is gonna give us this
area right over here.

00:03:44.518 --> 00:03:48.016
What's the area under the
curve negative x minus two,

00:03:48.016 --> 00:03:52.199
under that curve or under that
line and above the x-axis.

00:03:52.199 --> 00:03:55.372
And the second integral
is gonna give us this area

00:03:55.372 --> 00:03:57.489
right over here between x plus two

00:03:57.489 --> 00:04:01.406
and the x-axis going from
negative two to zero.

00:04:03.384 --> 00:04:05.661
And so let's evaluate each of these

00:04:05.661 --> 00:04:07.249
and you might even be able
to just evaluate these

00:04:07.249 --> 00:04:09.158
with a little bit of triangle areas,

00:04:09.158 --> 00:04:12.124
but let's just do this
analytically or algebraically.

00:04:12.124 --> 00:04:14.442
And so what's the
anti-derivative of negative x?

00:04:14.442 --> 00:04:17.696
Well that's negative x-squared over two,

00:04:17.696 --> 00:04:19.612
and then we have the negative two,

00:04:19.612 --> 00:04:21.444
so this is gonna be the anti-derivative

00:04:21.444 --> 00:04:23.274
is negative two x, we're
gonna evaluate that

00:04:23.274 --> 00:04:26.107
at negative two and negative four.

00:04:30.540 --> 00:04:33.188
And so that part is going to be what?

00:04:33.188 --> 00:04:36.158
Negative two squared, so it's the negative

00:04:36.158 --> 00:04:37.337
of negative two squared.

00:04:37.337 --> 00:04:41.279
So it's negative four over two
minus two times negative two.

00:04:41.279 --> 00:04:42.362
So plus four.

00:04:43.271 --> 00:04:46.315
So that's it evaluated at negative two.

00:04:46.315 --> 00:04:50.482
And then minus, if we
evaluate it at negative four.

00:04:52.700 --> 00:04:54.816
So we're gonna have minus
negative four squared

00:04:54.816 --> 00:04:59.039
is 16 over two, minus
two times negative four.

00:04:59.039 --> 00:05:00.872
So that is plus eight.

00:05:02.698 --> 00:05:05.793
So what is that going to give us?

00:05:05.793 --> 00:05:09.543
So this is negative two,
this right over here

00:05:11.239 --> 00:05:15.406
is negative eight, so the
second term right over here

00:05:17.903 --> 00:05:20.178
is just going to be equal to zero.

00:05:20.178 --> 00:05:21.115
Did I do that right?

00:05:21.115 --> 00:05:23.926
Yeah, the 16 over two, it's
negative and this is positive.

00:05:23.926 --> 00:05:26.075
Okay, so this is just going to be zero.

00:05:26.075 --> 00:05:28.965
And this is negative two plus four

00:05:28.965 --> 00:05:31.238
which is going to be equal to two.

00:05:31.238 --> 00:05:34.854
So what we have here in
magenta is equal to two.

00:05:34.854 --> 00:05:36.927
And what we have here in the blue,

00:05:36.927 --> 00:05:39.206
well let's see, this
is the anti-derivative

00:05:39.206 --> 00:05:42.039
of x-squared over two, plus two x,

00:05:44.697 --> 00:05:47.543
gonna evaluate it at
zero and negative two.

00:05:47.543 --> 00:05:48.928
You evaluate this thing at zero,

00:05:48.928 --> 00:05:50.674
it's just gonna be zero and from that

00:05:50.674 --> 00:05:54.416
you're going to subtract
negative two squared over two.

00:05:54.416 --> 00:05:57.545
That is positive four over two

00:05:57.545 --> 00:05:58.847
which is positive two.

00:05:58.847 --> 00:06:01.122
And then plus two times negative two.

00:06:01.122 --> 00:06:02.289
So minus four.

00:06:05.629 --> 00:06:09.659
And so this is going to be the negative

00:06:09.659 --> 00:06:12.409
of negative two, or positive two.

00:06:13.318 --> 00:06:14.539
So it's two plus two.

00:06:14.539 --> 00:06:16.088
And that makes sense that what we have

00:06:16.088 --> 00:06:19.135
in magenta here is two
and what we have over here

00:06:19.135 --> 00:06:21.968
is two, there's the symmetry here.

00:06:23.083 --> 00:06:25.273
There is a symmetry here.

00:06:25.273 --> 00:06:27.101
And so you add 'em all together

00:06:27.101 --> 00:06:31.700
and you get our integral is
going to be equal to four.

00:06:31.700 --> 00:06:34.147
And once again, just as a reality check

00:06:34.147 --> 00:06:35.936
you could say, look,
the height here is two,

00:06:35.936 --> 00:06:38.988
the width, the base here is two.

00:06:38.988 --> 00:06:42.160
Two times two times one-half
is indeed equal to two.

00:06:42.160 --> 00:06:43.706
Same thing over here.

00:06:43.706 --> 00:06:45.376
So that's the more geometric argument

00:06:45.376 --> 00:06:47.858
for why that area's two, that area is two,

00:06:47.858 --> 00:06:51.108
add 'em together you get positive four.


Definite integral of piecewise function

WEBVTT
Kind: captions
Language: en

00:00:00.315 --> 00:00:02.096
- [Voiceover] So we have
a f of x right over here

00:00:02.096 --> 00:00:04.860
and it's defined piecewise
for x less than zero,

00:00:04.860 --> 00:00:06.358
f of x is x plus one,

00:00:06.358 --> 00:00:08.535
for x is greater than or equal to zero,

00:00:08.535 --> 00:00:10.936
f of x is cosine of pi x.

00:00:10.936 --> 00:00:12.664
And we want to evaluate
the definite integral

00:00:12.664 --> 00:00:15.564
from negative one to one of f of x dx.

00:00:15.564 --> 00:00:16.397
And you might immediately say,

00:00:16.397 --> 00:00:19.647
well, which of these versions of f of x

00:00:20.598 --> 00:00:22.851
am I going to take the
antiderivative from,

00:00:22.851 --> 00:00:24.866
because from negative one to zero,

00:00:24.866 --> 00:00:27.213
I would think about x plus one,

00:00:27.213 --> 00:00:28.190
but then from zero to one

00:00:28.190 --> 00:00:30.747
I would think about cosine pi x.

00:00:30.747 --> 00:00:31.751
And if you were thinking that,

00:00:31.751 --> 00:00:33.205
you're thinking in the right direction.

00:00:33.205 --> 00:00:35.125
And the way that we can make this

00:00:35.125 --> 00:00:36.177
a little bit more straightforward

00:00:36.177 --> 00:00:38.866
is to actually split up
this definite integral.

00:00:38.866 --> 00:00:40.481
This is going to be equal
to the definite integral

00:00:40.481 --> 00:00:42.564
from negative one to zero

00:00:43.530 --> 00:00:44.780
of f of x dx

00:00:48.492 --> 00:00:49.740
plus the integral

00:00:49.740 --> 00:00:51.400
from zero to one

00:00:51.400 --> 00:00:52.733
of f of x dx.

00:00:53.797 --> 00:00:56.320
Now why was it useful for
me to split it up this way,

00:00:56.320 --> 00:00:58.413
in particular to split the integral

00:00:58.413 --> 00:00:59.658
from negative one to one,

00:00:59.658 --> 00:01:01.908
split it into two intervals
from negative one to zero,

00:01:01.908 --> 00:01:03.295
and zero to one?

00:01:03.295 --> 00:01:05.605
Well, I did that because x equals zero

00:01:05.605 --> 00:01:06.438
is where we switch,

00:01:06.438 --> 00:01:08.406
where f of x switches
from being x plus one

00:01:08.406 --> 00:01:09.656
to cosine pi x.

00:01:10.626 --> 00:01:13.490
So if you look at the interval
from negative one to zero,

00:01:13.490 --> 00:01:15.776
f of x is x plus one.

00:01:15.776 --> 00:01:19.415
So f of x here is x plus one.

00:01:19.415 --> 00:01:21.280
And then when you go from zero to one,

00:01:21.280 --> 00:01:23.363
f of x is cosine pi x.

00:01:24.402 --> 00:01:25.902
So cosine of pi x.

00:01:28.802 --> 00:01:31.576
And so now we just have to
evaluate each of these separately

00:01:31.576 --> 00:01:33.461
and add them together.

00:01:33.461 --> 00:01:36.365
So let's take the definite
integral from negative one

00:01:36.365 --> 00:01:38.698
to zero of x plus one dx.

00:01:41.093 --> 00:01:41.926
Well, let's see.

00:01:41.926 --> 00:01:44.512
The antiderivative x plus one is...

00:01:44.512 --> 00:01:48.455
antiderivative x is x squared over two.

00:01:48.455 --> 00:01:50.393
I'm just incrementing the exponent

00:01:50.393 --> 00:01:52.121
and then dividing by that value.

00:01:52.121 --> 00:01:53.999
And then plus x,

00:01:53.999 --> 00:01:55.768
and you could view it as
I'm doing the same thing.

00:01:55.768 --> 00:01:58.208
If this is x to the zero,
it'll be x to the first,

00:01:58.208 --> 00:02:00.018
x to the first over one,

00:02:00.018 --> 00:02:01.351
which is just x.

00:02:02.309 --> 00:02:05.219
And I'm gonna evaluate that at zero

00:02:05.219 --> 00:02:06.108
and subtract from that,

00:02:06.108 --> 00:02:07.581
it evaluated at one.

00:02:07.581 --> 00:02:09.667
Sorry, it evaluated at negative one.

00:02:09.667 --> 00:02:11.741
And so this is going to be equal to...

00:02:11.741 --> 00:02:13.824
If I evaluate it at zero,

00:02:14.866 --> 00:02:16.634
let me do this in another color.

00:02:16.634 --> 00:02:18.118
If I evaluate it at zero,

00:02:18.118 --> 00:02:21.501
it's going to be zero squared over two,

00:02:21.501 --> 00:02:22.992
which is, well, I'll just write it.

00:02:22.992 --> 00:02:26.136
Zero squared over two plus zero.

00:02:26.136 --> 00:02:30.136
Well, all of that's just
gonna be equal to zero.

00:02:31.817 --> 00:02:34.471
It evaluated at negative one.

00:02:34.471 --> 00:02:37.608
So minus negative one squared.

00:02:37.608 --> 00:02:41.719
Negative one squared over
two plus negative one.

00:02:41.719 --> 00:02:44.549
So negative one squared is just one.

00:02:44.549 --> 00:02:48.527
So it's 1/2 plus negative one.

00:02:48.527 --> 00:02:51.277
1/2 plus negative one,

00:02:52.320 --> 00:02:53.727
or 1/2 minus one,

00:02:53.727 --> 00:02:54.742
is negative 1/2.

00:02:54.742 --> 00:02:56.960
So all of that is negative 1/2.

00:02:56.960 --> 00:02:58.607
But then we're subtracting negative 1/2.

00:02:58.607 --> 00:03:00.828
Zero minus negative 1/2

00:03:00.828 --> 00:03:03.613
is going to be equal to positive 1/2.

00:03:03.613 --> 00:03:08.211
So this is going to be
equal to positive 1/2.

00:03:08.211 --> 00:03:10.364
So this first part right over here

00:03:10.364 --> 00:03:13.305
is positive 1/2.

00:03:13.305 --> 00:03:15.910
And now let's evaluate the integral

00:03:15.910 --> 00:03:18.410
from zero to one of cosine pi,

00:03:21.099 --> 00:03:22.731
I don't need that first parentheses,

00:03:22.731 --> 00:03:24.731
of cosine of pi x dx.

00:03:27.967 --> 00:03:29.774
What is this equal to?

00:03:29.774 --> 00:03:31.190
Now, if we were just trying to find

00:03:31.190 --> 00:03:33.092
the antiderivative of cosine of x,

00:03:33.092 --> 00:03:34.109
it's pretty straightforward.

00:03:34.109 --> 00:03:36.593
We know that the derivative
with respect to x

00:03:36.593 --> 00:03:38.184
of sine of x

00:03:38.184 --> 00:03:40.184
is equal to cosine of x.

00:03:44.104 --> 00:03:45.128
But that's not what we have here,

00:03:45.128 --> 00:03:47.490
we have cosine of pi x.

00:03:47.490 --> 00:03:49.297
So there is a technique here,

00:03:49.297 --> 00:03:50.675
you can call it u-substitution.

00:03:50.675 --> 00:03:52.131
You can say u is equal to pi x.

00:03:52.131 --> 00:03:52.964
If you don't know how to do that,

00:03:52.964 --> 00:03:55.319
you can still try to think this through,

00:03:55.319 --> 00:03:56.603
where we could say, alright,

00:03:56.603 --> 00:04:00.314
well, maybe it involves
sine of pi x somehow.

00:04:00.314 --> 00:04:02.308
So the derivative with respect to x

00:04:02.308 --> 00:04:03.558
of sine of pi x

00:04:06.025 --> 00:04:07.329
would be what?

00:04:07.329 --> 00:04:08.764
Well, we would use the chain rule.

00:04:08.764 --> 00:04:10.363
It would be the derivative
of the outside function

00:04:10.363 --> 00:04:12.187
with respect to the inside

00:04:12.187 --> 00:04:14.455
or sine of pi x with respect to pi x,

00:04:14.455 --> 00:04:16.955
which would be cosine of pi x,

00:04:18.085 --> 00:04:20.643
and then times the derivative
of the inside function

00:04:20.643 --> 00:04:22.142
with respect to x.

00:04:22.142 --> 00:04:24.377
So it would be times pi.

00:04:24.377 --> 00:04:26.493
Or you could say the
derivative of sine pi x

00:04:26.493 --> 00:04:28.243
is pi cosine of pi x.

00:04:30.465 --> 00:04:32.320
Now, we almost have that here,

00:04:32.320 --> 00:04:34.418
except we just need a pi.

00:04:34.418 --> 00:04:36.235
So what if we were to
throw a pi right over here,

00:04:36.235 --> 00:04:37.424
but so we don't change the value

00:04:37.424 --> 00:04:39.745
we also multiply by one over pi?

00:04:39.745 --> 00:04:42.474
So if you divide and
multiply by the same number,

00:04:42.474 --> 00:04:44.577
you're not changing its value.

00:04:44.577 --> 00:04:46.947
One over pi times pi is just equal to one.

00:04:46.947 --> 00:04:48.998
But this is useful.

00:04:48.998 --> 00:04:53.165
This is useful because we
now know that pi cosine pi x

00:04:54.374 --> 00:04:56.773
is the derivative of sine pi x.

00:04:56.773 --> 00:04:58.945
So this is all going to be equal to...

00:04:58.945 --> 00:05:00.695
This is equal to one.

00:05:01.947 --> 00:05:03.874
Let me take that one over pi.

00:05:03.874 --> 00:05:06.404
So this is equal to one over pi

00:05:06.404 --> 00:05:07.237
times...

00:05:08.332 --> 00:05:10.080
Now we're going to evaluate.

00:05:10.080 --> 00:05:11.747
So the antiderivative here we just said

00:05:11.747 --> 00:05:13.080
is sine of pi x,

00:05:17.972 --> 00:05:22.139
and we're going to evaluate
that at one and at zero.

00:05:23.553 --> 00:05:27.900
So this is going to be
equal to one over pi.

00:05:27.900 --> 00:05:29.831
One over pi, not pi.

00:05:29.831 --> 00:05:31.819
My hand is not listening to my mouth.

00:05:31.819 --> 00:05:34.152
One over pi times sine of pi

00:05:41.921 --> 00:05:46.428
minus sine of pi times
zero, which is just zero.

00:05:46.428 --> 00:05:49.379
Well, sine of pi, that's zero.

00:05:49.379 --> 00:05:51.431
Sine of zero is zero.

00:05:51.431 --> 00:05:52.447
So you're gonna have one over pi

00:05:52.447 --> 00:05:53.917
times zero minus zero.

00:05:53.917 --> 00:05:56.869
So this whole thing is just
all going to be equal to zero.

00:05:56.869 --> 00:05:59.278
So this first part was 1/2,

00:05:59.278 --> 00:06:02.563
this second part right
over here is equal to zero,

00:06:02.563 --> 00:06:05.953
so the whole definite integral
is gonna be 1/2 plus zero,

00:06:05.953 --> 00:06:08.797
which is equal to 1/2.

00:06:08.797 --> 00:06:12.862
So all of that together is equal to 1/2.


Definite integral of rational function

WEBVTT
Kind: captions
Language: en

00:00:00.491 --> 00:00:02.325
- [Voiceover] So we wanna
evaluate the definite integral

00:00:02.325 --> 00:00:05.075
from negative one to negative two

00:00:06.008 --> 00:00:10.920
of 16 minus x to the third
over x to the third dx.

00:00:10.920 --> 00:00:12.749
Now at first this might seem daunting,

00:00:12.749 --> 00:00:15.476
I have this rational expression,
I have xs in the numerators

00:00:15.476 --> 00:00:18.076
and xs in the denominators,
but we just have to remember,

00:00:18.076 --> 00:00:20.317
we just have to do some
algebraic manipulation,

00:00:20.317 --> 00:00:22.571
and this is going to seem
a lot more attractable.

00:00:22.571 --> 00:00:24.901
This is the same thing
as the definite integral

00:00:24.901 --> 00:00:28.596
from negative one to negative two of

00:00:28.596 --> 00:00:30.679
16 over x to the third

00:00:31.636 --> 00:00:33.545
minus x to the third

00:00:33.545 --> 00:00:35.175
over x to the third,

00:00:35.175 --> 00:00:38.954
minus x to the third
over x to the third dx.

00:00:38.954 --> 00:00:41.421
And now what is that going to be equal to?

00:00:41.421 --> 00:00:44.404
That is going be equal
to the definite integral

00:00:44.404 --> 00:00:47.487
from negative one to negative two of,

00:00:48.618 --> 00:00:52.523
I could write this first
term right over here,

00:00:52.523 --> 00:00:54.292
let me do this in a different color.

00:00:54.292 --> 00:00:57.646
I could write, I could write this

00:00:57.646 --> 00:01:01.813
as 16x to the negative three,
x to the negative three,

00:01:02.716 --> 00:01:05.130
and this second one,

00:01:05.130 --> 00:01:09.939
we have minus x to the
third over x to the third.

00:01:09.939 --> 00:01:11.235
Well x to the third is just over,

00:01:11.235 --> 00:01:12.556
x to the third over x to the third

00:01:12.556 --> 00:01:14.843
is just going to be equal to one.

00:01:14.843 --> 00:01:17.843
So this is going to be minus one dx,

00:01:19.893 --> 00:01:20.726
so dx.

00:01:21.842 --> 00:01:23.934
And so what is this going to be equal to?

00:01:23.934 --> 00:01:26.934
Well, let's take the antiderivative
of each of these parts

00:01:26.934 --> 00:01:28.043
and then we're going to
have to evaluate them

00:01:28.043 --> 00:01:29.943
at the different bounds.

00:01:29.943 --> 00:01:31.030
So, let's see.

00:01:31.030 --> 00:01:34.210
The antiderivative of 16x
to the negative three,

00:01:34.210 --> 00:01:35.552
we're just gonna do the power rule

00:01:35.552 --> 00:01:37.151
for derivatives in reverse.

00:01:37.151 --> 00:01:39.465
You can view this as the
power rule of integration

00:01:39.465 --> 00:01:42.286
or the power rule of
taking the antiderivative

00:01:42.286 --> 00:01:43.576
where what you do is

00:01:43.576 --> 00:01:46.491
you're gonna increase our exponent by one,

00:01:46.491 --> 00:01:49.515
so you're gonna go from
negative three to negative two,

00:01:49.515 --> 00:01:52.565
and then you're gonna
divide by that amount,

00:01:52.565 --> 00:01:53.461
by negative two.

00:01:53.461 --> 00:01:57.670
So it's going to be 16
divided by negative two

00:01:57.670 --> 00:01:59.268
times x to the negative two.

00:01:59.268 --> 00:02:01.008
All I did is I increased the exponent

00:02:01.008 --> 00:02:02.980
and I divided by that amount.

00:02:02.980 --> 00:02:04.727
So that's the antiderivative here.

00:02:04.727 --> 00:02:07.695
And 16 divided by negative two,

00:02:07.695 --> 00:02:10.072
that is just negative eight.

00:02:10.072 --> 00:02:12.401
So we have negative eight,
x to the negative two.

00:02:12.401 --> 00:02:15.343
And then the antiderivative
of negative one,

00:02:15.343 --> 00:02:19.510
well, that's just negative
x, negative, negative x,

00:02:21.654 --> 00:02:22.571
negative x.

00:02:23.537 --> 00:02:25.648
And actually you might just know that,

00:02:25.648 --> 00:02:27.653
and then hey, if I take the
derivative of negative x

00:02:27.653 --> 00:02:29.751
I get negative one, or if you view this

00:02:29.751 --> 00:02:33.421
as negative x to the zero power,
because that's what one is,

00:02:33.421 --> 00:02:34.838
well, it's the same thing.

00:02:34.838 --> 00:02:37.036
You increase the exponent by one

00:02:37.036 --> 00:02:39.033
to get x to the first power.

00:02:39.033 --> 00:02:40.831
And then you divide by one and so,

00:02:40.831 --> 00:02:43.516
I mean, you can view it
as that right over there,

00:02:43.516 --> 00:02:48.177
but either way you get
to negative or minus x.

00:02:48.177 --> 00:02:50.898
And so now we want to evaluate that.

00:02:50.898 --> 00:02:52.935
We're going to evaluate that at the bounds

00:02:52.935 --> 00:02:54.297
and take the difference.

00:02:54.297 --> 00:02:57.063
So we're going to evaluate
that at negative two

00:02:57.063 --> 00:03:01.461
and then subtract from that
this evaluated at negative one.

00:03:01.461 --> 00:03:03.214
And let me do those in
two different colors

00:03:03.214 --> 00:03:04.966
so we can see what's going on.

00:03:04.966 --> 00:03:06.855
So we're gonna evaluate it at negative two

00:03:06.855 --> 00:03:09.765
and we're gonna evaluate
it at negative one.

00:03:09.765 --> 00:03:12.277
So let's first evaluate
it at negative two.

00:03:12.277 --> 00:03:14.789
So this is going to be equal to,

00:03:14.789 --> 00:03:16.554
this is going to be equal to,

00:03:16.554 --> 00:03:18.328
when you evaluate it at negative two,

00:03:18.328 --> 00:03:22.683
it's going to be negative
eight, negative eight

00:03:22.683 --> 00:03:24.842
times x to the negative two.

00:03:24.842 --> 00:03:27.913
So negative two the the negative two power

00:03:27.913 --> 00:03:29.496
minus negative two.

00:03:30.624 --> 00:03:33.142
And from that we're going to subtract it

00:03:33.142 --> 00:03:35.817
and evaluate it at negative one.

00:03:35.817 --> 00:03:40.028
So, it's going to be negative
eight times negative one

00:03:40.028 --> 00:03:43.778
to the negative two
power minus negative one.

00:03:45.000 --> 00:03:47.655
All right, so then what
is this going to be?

00:03:47.655 --> 00:03:50.677
So, negative two the negative two.

00:03:50.677 --> 00:03:53.143
So negative two to the negative two

00:03:53.143 --> 00:03:57.356
is equal to one over negative two squared,

00:03:57.356 --> 00:03:59.962
which is equal to 1/4.

00:03:59.962 --> 00:04:03.317
So this is equal to positive 1/4,

00:04:03.317 --> 00:04:07.330
but then negative eight times positive 1/4

00:04:07.330 --> 00:04:10.413
is going to be equal to negative two.

00:04:11.731 --> 00:04:14.611
And then we have negative
two minus negative two,

00:04:14.611 --> 00:04:17.417
so that's negative two plus two.

00:04:17.417 --> 00:04:20.431
And so everything I've just
done in this purplish color

00:04:20.431 --> 00:04:22.664
that is just going to be zero.

00:04:22.664 --> 00:04:26.545
And then if we look at what's
going on in the orange,

00:04:26.545 --> 00:04:28.061
when you evaluate it at negative one,

00:04:28.061 --> 00:04:31.208
let's see, negative one
to the negative two power,

00:04:31.208 --> 00:04:33.116
well, that's one over
negative one squared.

00:04:33.116 --> 00:04:34.884
Well, this all is just going to be one.

00:04:34.884 --> 00:04:39.151
And so we're gonna have
negative eight plus one,

00:04:39.151 --> 00:04:41.901
which is equal to negative seven.

00:04:43.479 --> 00:04:46.109
So all of this evaluates
to negative seven.

00:04:46.109 --> 00:04:48.767
But remember, we're
subtracting negative seven,

00:04:48.767 --> 00:04:50.477
and so this is going to result,

00:04:50.477 --> 00:04:51.836
we deserve a little bit of a drum roll,

00:04:51.836 --> 00:04:56.402
this is going to be equal
to positive, positive seven.

00:04:56.402 --> 00:04:57.235
And obviously you don't have to write

00:04:57.235 --> 00:04:58.119
that positive out in front,

00:04:58.119 --> 00:05:00.447
I just wrote that just to emphasize

00:05:00.447 --> 00:05:03.876
that this is going to be a positive seven.


Worked examples: Definite integral properties 1

WEBVTT
Kind: captions
Language: en

00:00:00.121 --> 00:00:02.110
- Do you want to evaluate
the definite integral

00:00:02.110 --> 00:00:05.624
from three to three, of F of X, D X.

00:00:05.624 --> 00:00:07.733
And we're given the graph of F of X,

00:00:07.733 --> 00:00:09.681
and of Y equals F of X,

00:00:09.681 --> 00:00:12.077
and the area between F of X,

00:00:12.077 --> 00:00:15.747
and the X-axis over different intervals.

00:00:15.747 --> 00:00:17.218
Well when you look at this,

00:00:17.218 --> 00:00:18.368
you actually don't even have to look

00:00:18.368 --> 00:00:19.507
at this graph over here,

00:00:19.507 --> 00:00:22.030
because in general, if I
have the definite integral

00:00:22.030 --> 00:00:25.135
of any function F of X, D X,

00:00:25.135 --> 00:00:27.933
from let's say, A to the same value,

00:00:27.933 --> 00:00:29.320
from one value to the same value,

00:00:29.320 --> 00:00:32.928
this is always going to be equal to zero.

00:00:32.928 --> 00:00:34.410
We're going from three to three.

00:00:34.410 --> 00:00:36.592
We could be going from
negative pi to negative pi.

00:00:36.592 --> 00:00:38.224
It's always going to be zero.

00:00:38.224 --> 00:00:39.057
One way to think about it,

00:00:39.057 --> 00:00:40.747
we're starting and stopping here at three,

00:00:40.747 --> 00:00:43.614
so we're not capturing any area.

00:00:43.614 --> 00:00:45.399
Let's do another one.

00:00:45.399 --> 00:00:49.102
So here, we want to find
the definite integral

00:00:49.102 --> 00:00:52.019
from seven to four, of F of X, D X.

00:00:55.421 --> 00:00:58.178
So we want to go from seven to four.

00:00:58.178 --> 00:01:01.245
So we want to go from seven to four.

00:01:01.245 --> 00:01:02.642
Now you might be tempted to say,

00:01:02.642 --> 00:01:06.689
okay well look, the area
between F of X and X, is two,

00:01:06.689 --> 00:01:09.039
so maybe this thing is two.

00:01:09.039 --> 00:01:12.370
But the key realization
is, this area only applies

00:01:12.370 --> 00:01:15.923
when you have the lower
bound as the lower bound,

00:01:15.923 --> 00:01:19.576
and the higher value as the higher bound.

00:01:19.576 --> 00:01:22.872
So the integral from four to seven

00:01:22.872 --> 00:01:25.122
of F of X, D X, this thing.

00:01:26.865 --> 00:01:28.856
This thing is equal to two.

00:01:28.856 --> 00:01:32.547
This thing is depicting
that area right over there.

00:01:32.547 --> 00:01:34.434
So what about this,
where we've switched it?

00:01:34.434 --> 00:01:35.598
Instead of going from four to seven,

00:01:35.598 --> 00:01:37.125
we're going from seven to four.

00:01:37.125 --> 00:01:40.567
Well the key realization is,
is if you switch the bounds,

00:01:40.567 --> 00:01:43.076
this is a key definite integral property,

00:01:43.076 --> 00:01:45.667
that's going to give
you the negative value.

00:01:45.667 --> 00:01:48.593
So this is going to be equal
to the negative of the integral

00:01:48.593 --> 00:01:51.510
from four to seven, of F of X, D X.

00:01:54.286 --> 00:01:57.029
And so this is going to be negative,

00:01:57.029 --> 00:01:59.372
and we just figured out the
integral from four to seven

00:01:59.372 --> 00:02:02.041
of F of X, D X, well
now that is this area.

00:02:02.041 --> 00:02:03.846
F of X is above the X-axis.

00:02:03.846 --> 00:02:05.122
It's a positive area.

00:02:05.122 --> 00:02:07.128
So it is going to be,

00:02:07.128 --> 00:02:10.074
so this thing right over
here is going to evaluate

00:02:10.074 --> 00:02:12.921
to positive two, but we have
that negative out front,

00:02:12.921 --> 00:02:17.088
so our original expression
would evaluate to negative two.


Worked examples: Definite integral properties 2

WEBVTT
Kind: captions
Language: en

00:00:00.571 --> 00:00:01.650
- [Voiceover] So what we're
gonna do in this video

00:00:01.650 --> 00:00:04.943
is several examples where
we evaluate expressions

00:00:04.943 --> 00:00:06.306
with definite integrals

00:00:06.306 --> 00:00:09.214
and so right over here we
have the definite integral

00:00:09.214 --> 00:00:11.160
from negative two to three of 2 f(x)dx

00:00:11.160 --> 00:00:14.401
plus the definite integral
from 3 to seven of

00:00:14.401 --> 00:00:15.486
3 f(x)dx

00:00:15.486 --> 00:00:16.626
and all we know about f(x)

00:00:16.626 --> 00:00:19.453
is the graph of y=f(x)

00:00:19.453 --> 00:00:22.005
from negative, from x equals negative six

00:00:22.005 --> 00:00:23.414
to x equals seven.

00:00:23.414 --> 00:00:25.179
They also give us the areas between

00:00:25.179 --> 00:00:27.560
f(x), y equals f(x)

00:00:27.560 --> 00:00:28.651
and the x axis.

00:00:28.651 --> 00:00:30.434
The negative areas show that our function

00:00:30.434 --> 00:00:32.635
is below the x axis

00:00:32.635 --> 00:00:33.642
and so given that,

00:00:33.642 --> 00:00:34.912
can we evaluate that

00:00:34.912 --> 00:00:36.048
and like always,

00:00:36.048 --> 00:00:38.002
pause the video and see if
you can do it on your own.

00:00:38.002 --> 00:00:40.524
Well the first thing that
my brain wants to do is

00:00:40.524 --> 00:00:41.357
I want to,

00:00:43.148 --> 00:00:46.161
I want to take these
constants out of the integral

00:00:46.161 --> 00:00:47.364
because then once they're out

00:00:47.364 --> 00:00:49.789
and I'm just taking the
straight up definite integrals

00:00:49.789 --> 00:00:50.813
of f(x),

00:00:50.813 --> 00:00:53.361
I can relate that to the areas over here

00:00:53.361 --> 00:00:54.538
and I know I can do that

00:00:54.538 --> 00:00:57.240
and this is a very common
integration property

00:00:57.240 --> 00:00:59.615
and applies to definite
and indefinite integrals

00:00:59.615 --> 00:01:01.418
but if I'm taking the integral of

00:01:01.418 --> 00:01:03.585
k f(x)dx,

00:01:05.136 --> 00:01:09.303
this is the same thing as k
times the integral of f(x)dx.

00:01:10.457 --> 00:01:12.398
So let's just apply that property there

00:01:12.398 --> 00:01:14.422
which is really you're
taking the scaler outside

00:01:14.422 --> 00:01:15.929
of the integral.

00:01:15.929 --> 00:01:18.182
To say this is going to be the same thing

00:01:18.182 --> 00:01:20.668
as two times the definite
integral from negative two

00:01:20.668 --> 00:01:24.001
to three of f(x)dx plus

00:01:26.505 --> 00:01:30.672
three times the integral from
three to seven of f(x)dx.

00:01:34.102 --> 00:01:35.073
All right.

00:01:35.073 --> 00:01:37.130
Now can we evaluate these things?

00:01:37.130 --> 00:01:39.463
So what is this going to be?

00:01:40.429 --> 00:01:42.860
The definite integral
from negative two to three

00:01:42.860 --> 00:01:43.918
of f(x)dx.

00:01:43.918 --> 00:01:45.655
Well we can view that as the area

00:01:45.655 --> 00:01:48.632
between the curves y
equals f(x) and the x axis,

00:01:48.632 --> 00:01:51.705
between x equals negative
two and x equals three.

00:01:51.705 --> 00:01:53.872
So between x equals two

00:01:54.986 --> 00:01:56.981
and x equals three,

00:01:56.981 --> 00:01:59.997
they give us the area
between y equals f(x)

00:01:59.997 --> 00:02:01.304
and the x axis.

00:02:01.304 --> 00:02:02.332
It is seven.

00:02:02.332 --> 00:02:03.573
So this thing over here

00:02:03.573 --> 00:02:04.701
is seven

00:02:04.701 --> 00:02:08.800
and then we have the
integral from three to seven

00:02:08.800 --> 00:02:10.116
of f(x)

00:02:10.116 --> 00:02:12.587
so we're gonna go from three to seven

00:02:12.587 --> 00:02:13.832
and once again,

00:02:13.832 --> 00:02:15.778
well this is going to evaluate

00:02:15.778 --> 00:02:17.134
to a negative value

00:02:17.134 --> 00:02:19.291
because f(x) is below the x axis there

00:02:19.291 --> 00:02:21.810
and it's going to evaluate
to negative three.

00:02:21.810 --> 00:02:24.708
So this is all going to be

00:02:24.708 --> 00:02:28.625
this is going to be two
times seven so 14 plus,

00:02:30.391 --> 00:02:33.224
14 plus three times negative three

00:02:34.990 --> 00:02:36.731
so plus negative nine

00:02:36.731 --> 00:02:39.898
and so 14 minus nine is equal to five.

00:02:41.012 --> 00:02:42.147
This is fun.

00:02:42.147 --> 00:02:43.385
Let's do more of these.

00:02:43.385 --> 00:02:44.519
All right.

00:02:44.519 --> 00:02:47.269
Okay, so here this first integral

00:02:49.705 --> 00:02:51.866
the integral from zero to five

00:02:51.866 --> 00:02:53.429
of f(x)dx,

00:02:53.429 --> 00:02:55.046
so this is pretty straightforward.

00:02:55.046 --> 00:02:56.531
This right over here,

00:02:56.531 --> 00:02:58.237
we're gonna go from zero to five.

00:02:58.237 --> 00:03:00.038
Zero to five

00:03:00.038 --> 00:03:02.031
and of f(x)dx,

00:03:02.031 --> 00:03:04.170
so we're talking about this area there

00:03:04.170 --> 00:03:05.643
which they tell us is four

00:03:05.643 --> 00:03:07.792
so that was pretty easy to evaluate

00:03:07.792 --> 00:03:09.596
and now we're gonna subtract,

00:03:09.596 --> 00:03:12.752
we're gonna subtract
going from negative eight

00:03:12.752 --> 00:03:16.315
to negative four times 2f(x),

00:03:16.315 --> 00:03:18.096
well let's just take this two outside.

00:03:18.096 --> 00:03:21.096
So if we just take this two outside,

00:03:22.303 --> 00:03:25.170
then this just becomes the
integral from negative eight

00:03:25.170 --> 00:03:27.174
to negative four of f(x)

00:03:27.174 --> 00:03:28.674
and so this thing,

00:03:30.024 --> 00:03:31.916
this thing right over here

00:03:31.916 --> 00:03:33.416
evaluates to five.

00:03:34.400 --> 00:03:36.627
It's this area they're talking about.

00:03:36.627 --> 00:03:38.232
So this is all going to simplify to

00:03:38.232 --> 00:03:41.417
four minus that two that we brought out,

00:03:41.417 --> 00:03:43.167
minus two times five.

00:03:44.763 --> 00:03:47.180
Times five, which is equal to

00:03:48.633 --> 00:03:50.550
let's see four minus 10

00:03:52.032 --> 00:03:54.997
which is equal to negative six.

00:03:54.997 --> 00:03:55.971
All right.

00:03:55.971 --> 00:03:58.436
Let's do another one of these.

00:03:58.436 --> 00:04:02.352
So here I have the integral
from negative seven

00:04:02.352 --> 00:04:03.689
to negative five

00:04:03.689 --> 00:04:05.483
so I'm going from negative seven

00:04:05.483 --> 00:04:07.535
to negative five

00:04:07.535 --> 00:04:10.053
which is, it's gonna be right around there

00:04:10.053 --> 00:04:11.366
so I want to find,

00:04:11.366 --> 00:04:13.366
I want to find this area

00:04:14.660 --> 00:04:16.327
right over here.

00:04:16.327 --> 00:04:18.075
So I want to find that area.

00:04:18.075 --> 00:04:19.695
That's that

00:04:19.695 --> 00:04:23.390
and then I'm gonna go from
negative five to zero.

00:04:23.390 --> 00:04:26.364
So then this is going to be

00:04:26.364 --> 00:04:29.031
going from negative five to zero

00:04:30.041 --> 00:04:32.458
so it's gonna be all of that.

00:04:33.400 --> 00:04:34.935
Now there's a couple of
ways you can think about

00:04:34.935 --> 00:04:35.979
doing it.

00:04:35.979 --> 00:04:37.697
You can assume I have some symmetry here

00:04:37.697 --> 00:04:39.254
and they don't tell it for sure

00:04:39.254 --> 00:04:42.345
but it looks very symmetric
around x equals negative five

00:04:42.345 --> 00:04:44.089
so you could assume that this eight

00:04:44.089 --> 00:04:45.576
is split between these two regions

00:04:45.576 --> 00:04:46.778
but an easier way to do it

00:04:46.778 --> 00:04:47.952
is just to realize look,

00:04:47.952 --> 00:04:49.951
I went from negative seven

00:04:49.951 --> 00:04:50.993
to negative five

00:04:50.993 --> 00:04:52.819
and then from negative five to zero

00:04:52.819 --> 00:04:54.366
and I'm integrating the same thing.

00:04:54.366 --> 00:04:55.722
F(x)dx

00:04:55.722 --> 00:04:57.673
so this integral I can rewrite

00:04:57.673 --> 00:05:00.898
as the integral from negative seven

00:05:00.898 --> 00:05:04.051
so negative seven all the way to zero

00:05:04.051 --> 00:05:07.871
of f(x), f(x)dx

00:05:07.871 --> 00:05:09.525
and so really that's just going to be

00:05:09.525 --> 00:05:11.901
the net area between negative seven

00:05:11.901 --> 00:05:13.067
and zero

00:05:13.067 --> 00:05:15.334
and so we have the positive eight there

00:05:15.334 --> 00:05:18.354
so this is going to be
equal to the positive eight

00:05:18.354 --> 00:05:20.719
and then we have the negative one there

00:05:20.719 --> 00:05:21.958
so minus one

00:05:21.958 --> 00:05:23.958
which is equal to seven.


Finding specific antiderivatives: rational function

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:03.509
- [Voiceover] So we're told
that F of two is equal to 12,

00:00:03.509 --> 00:00:08.037
F prime of X is equal to
24 over X to the third

00:00:08.037 --> 00:00:13.030
and what we want to figure out
is what is F of negative one.

00:00:13.030 --> 00:00:16.791
Alright, so they give us the
derivative in terms of X.

00:00:16.791 --> 00:00:18.927
So maybe we could take the anti-derivative

00:00:18.927 --> 00:00:21.899
of the derivative to find
our original function.

00:00:21.899 --> 00:00:23.100
So let's do that.

00:00:23.100 --> 00:00:25.848
So we could say that F of X,

00:00:25.848 --> 00:00:29.907
F of X is going to be equal
to the anti-derivative

00:00:29.907 --> 00:00:34.233
or we could say the indefinite
integral of F prime of X,

00:00:34.233 --> 00:00:37.945
which is equal to 24 over X to the third.

00:00:37.945 --> 00:00:41.283
I could write it over like
this, 24 over X to the third.

00:00:41.283 --> 00:00:44.300
But to help me process
it a little bit more,

00:00:44.300 --> 00:00:48.439
I'm gonna write this as
24 X to the negative three

00:00:48.439 --> 00:00:51.563
'cause then it'll become how
to take that anti-derivative.

00:00:51.563 --> 00:00:52.396
D, DX.

00:00:53.993 --> 00:00:55.809
And so what is the anti-derivative

00:00:55.809 --> 00:00:58.226
of 24X to the negative three?

00:00:59.334 --> 00:01:03.767
Well we're just going to do
the power rule in reverse.

00:01:03.767 --> 00:01:05.262
So what we're going to do

00:01:05.262 --> 00:01:08.814
is we're going to increase the exponent.

00:01:08.814 --> 00:01:09.908
So let me just rewrite it.

00:01:09.908 --> 00:01:12.445
It's going to be 24X to the,

00:01:12.445 --> 00:01:14.844
we're going to increase
the exponent by one,

00:01:14.844 --> 00:01:17.809
so it's gonna be X to the
negative three plus one

00:01:17.809 --> 00:01:20.372
and then we're going to divide
by that increased exponent.

00:01:20.372 --> 00:01:23.149
So negative three plus one

00:01:23.149 --> 00:01:25.606
and so that is going to be

00:01:25.606 --> 00:01:29.317
negative three plus one
is X to the negative two

00:01:29.317 --> 00:01:32.148
and then we divide by negative two

00:01:32.148 --> 00:01:34.658
and if you're in doubt
about what we just did

00:01:34.658 --> 00:01:37.168
where we're kind of doing
the power rule in reverse,

00:01:37.168 --> 00:01:40.079
now take the derivative of
this using the power rule.

00:01:40.079 --> 00:01:43.417
Negative two times 24 over negative two

00:01:43.417 --> 00:01:46.274
is just gonna be 24 and then
you decrement that exponent,

00:01:46.274 --> 00:01:47.849
you go to negative three.

00:01:47.849 --> 00:01:50.760
So are we done here, is this F of X?

00:01:50.760 --> 00:01:54.098
Well F of X might involve a constant.

00:01:54.098 --> 00:01:57.302
So let's put a constant out here

00:01:57.302 --> 00:01:58.366
because notice if you were to take

00:01:58.366 --> 00:02:00.289
the derivative of this thing here,

00:02:00.289 --> 00:02:03.013
the derivative of 24X to the
negative two over negative two

00:02:03.013 --> 00:02:06.217
we already established is
24X to the negative three.

00:02:06.217 --> 00:02:07.735
But then if you take the
derivative of a constant,

00:02:07.735 --> 00:02:08.777
well that just disappears.

00:02:08.777 --> 00:02:10.966
So you don't see it when
you look at the derivative.

00:02:10.966 --> 00:02:13.423
So we have to make sure that
there might be a constant

00:02:13.423 --> 00:02:15.079
and I have a feeling based
on the information that

00:02:15.079 --> 00:02:17.989
they've given us, we're going
to make use of that constant.

00:02:17.989 --> 00:02:20.152
So let me rewrite F of X.

00:02:20.152 --> 00:02:23.917
So we know that F of X can be expressed as

00:02:23.917 --> 00:02:25.493
24 divided by negative two.

00:02:25.493 --> 00:02:29.660
It's negative 12X to the
negative two plus some constant.

00:02:31.675 --> 00:02:34.002
So how do we figure out that constant?

00:02:34.002 --> 00:02:36.592
Well they have told us what F of two is.

00:02:36.592 --> 00:02:38.728
F of two is equal to 12.

00:02:38.728 --> 00:02:40.117
So let's write this down.

00:02:40.117 --> 00:02:42.367
So F of two is equal to 12,

00:02:47.407 --> 00:02:49.543
which is equal to well we
just have to put two in

00:02:49.543 --> 00:02:51.092
everywhere we see an X.

00:02:51.092 --> 00:02:52.788
That's going to be negative two

00:02:52.788 --> 00:02:56.288
times two to the negative two power plus C

00:02:58.235 --> 00:03:01.609
and so 12 is equal to, what is this,

00:03:01.609 --> 00:03:03.038
two to the negative two.

00:03:03.038 --> 00:03:07.017
Two to the negative two is
equal to one over two squared,

00:03:07.017 --> 00:03:09.206
which is equal to 1/4.

00:03:09.206 --> 00:03:13.212
So this is negative 12 times 1/4.

00:03:13.212 --> 00:03:15.855
Negative 12 times 1/4 is negative three.

00:03:15.855 --> 00:03:18.739
So it's negative three plus C.

00:03:18.739 --> 00:03:21.436
Now we can add three to
both sides to solve or C.

00:03:21.436 --> 00:03:23.436
We get 15 is equal to C.

00:03:27.364 --> 00:03:29.281
So or C is equal to 15.

00:03:31.236 --> 00:03:32.892
That is equal to 15.

00:03:32.892 --> 00:03:35.081
And so now we can write our F of X

00:03:35.081 --> 00:03:38.414
as we get F of X is equal to negative 12

00:03:42.932 --> 00:03:43.765
and I could even write that as

00:03:43.765 --> 00:03:46.747
negative 12 over X squared if we like.

00:03:46.747 --> 00:03:49.580
Negative 12 over X squared plus 15

00:03:52.568 --> 00:03:56.066
and now using that we can
evaluate F of negative one.

00:03:56.066 --> 00:03:57.374
F of negative one.

00:03:57.374 --> 00:04:00.098
Everywhere we see an X we
put a negative one there.

00:04:00.098 --> 00:04:03.383
So this is going to be
negative one squared.

00:04:03.383 --> 00:04:07.300
So F of negative one is
equal to 12 divided by,

00:04:08.377 --> 00:04:10.673
negative 12 divided by
negative one squared.

00:04:10.673 --> 00:04:11.955
Well negative one squared is just one.

00:04:11.955 --> 00:04:15.627
So it's gonna be negative 12 plus 15,

00:04:15.627 --> 00:04:18.877
which is equal to three and we're done.

00:04:19.846 --> 00:04:22.263
This thing is equal to three.


Finding specific antiderivatives: exponential function

WEBVTT
Kind: captions
Language: en

00:00:00.667 --> 00:00:02.040
- [Voiceover] We're told that F of seven

00:00:02.040 --> 00:00:06.165
is equal to 40 plus five,
E to the seventh power,

00:00:06.165 --> 00:00:08.234
and F prime of X is equal to five,

00:00:08.234 --> 00:00:09.263
E to the X.

00:00:09.263 --> 00:00:11.420
What is F of zero?

00:00:11.420 --> 00:00:12.810
So to evaluate F of zero,

00:00:12.810 --> 00:00:15.980
let's take the anti-derivative
of F prime of X,

00:00:15.980 --> 00:00:18.109
and then we're going to have a constant

00:00:18.109 --> 00:00:19.293
of integration there,

00:00:19.293 --> 00:00:21.892
so we can use the information
that they gave us up here

00:00:21.892 --> 00:00:23.974
that F of seven is equal to this.

00:00:23.974 --> 00:00:25.661
This might look like an expression.

00:00:25.661 --> 00:00:26.869
Well, it is an expression,

00:00:26.869 --> 00:00:28.060
but it's really just a number.

00:00:28.060 --> 00:00:29.717
There's no variables in this,

00:00:29.717 --> 00:00:31.171
and so we can use that to solve

00:00:31.171 --> 00:00:32.696
for our constant of integration,

00:00:32.696 --> 00:00:34.085
and then we will have fully known

00:00:34.085 --> 00:00:35.120
what F of X is,

00:00:35.120 --> 00:00:36.838
and we can use that to evaluate F of zero,

00:00:36.838 --> 00:00:38.229
so let's just do it.

00:00:38.229 --> 00:00:40.856
So if F prime of X is
equal to five, E to the X,

00:00:40.856 --> 00:00:43.893
then F of X is going to be equal

00:00:43.893 --> 00:00:46.520
to the anti-derivative of F prime of X,

00:00:46.520 --> 00:00:50.437
or the anti-derivative
of five, E to the X, DX,

00:00:51.562 --> 00:00:53.517
and this is the thing
that I always find amazing

00:00:53.517 --> 00:00:54.396
about exponentials,

00:00:54.396 --> 00:00:55.540
and actually, let me just take a step.

00:00:55.540 --> 00:00:57.437
I'll take that five out of the integral

00:00:57.437 --> 00:00:59.844
so it becomes a little bit more obvious.

00:00:59.844 --> 00:01:01.489
And so the anti-derivative of E to the X,

00:01:01.489 --> 00:01:03.056
well, that's just E to the X

00:01:03.056 --> 00:01:04.396
because the derivative of E to the X

00:01:04.396 --> 00:01:05.229
is E to the X,

00:01:05.229 --> 00:01:06.340
which I find amazing every time

00:01:06.340 --> 00:01:08.318
I have to manipulate
or take the derivative

00:01:08.318 --> 00:01:10.159
or anti-derivative of E to the X.

00:01:10.159 --> 00:01:13.909
So this is gonna be
five, E to the X, plus C,

00:01:15.302 --> 00:01:16.139
and you can verify.

00:01:16.139 --> 00:01:19.223
Take the derivative of
five, E to the X, plus C.

00:01:19.223 --> 00:01:21.031
The derivative of five, E to the X,

00:01:21.031 --> 00:01:22.992
well, that's five, E to the X,

00:01:22.992 --> 00:01:24.299
so that works out.

00:01:24.299 --> 00:01:26.227
Well, and the derivative of C is zero,

00:01:26.227 --> 00:01:27.950
so you wouldn't see it over here.

00:01:27.950 --> 00:01:30.783
So now, let's use this information

00:01:32.362 --> 00:01:34.298
to figure out what C is

00:01:34.298 --> 00:01:36.361
so that we know exactly what F of X is,

00:01:36.361 --> 00:01:38.319
and then we can evaluate F of zero.

00:01:38.319 --> 00:01:40.081
So we know that F of seven,

00:01:40.081 --> 00:01:42.057
so when X is equal to seven,

00:01:42.057 --> 00:01:46.197
this expression is going
to evaluate to this thing,

00:01:46.197 --> 00:01:47.670
40 plus five, E to the seven.

00:01:47.670 --> 00:01:50.753
So, five times E to the seventh power

00:01:54.322 --> 00:01:57.072
plus C is equal to 40, plus five,

00:02:02.142 --> 00:02:04.059
E to the seventh power.

00:02:04.973 --> 00:02:06.232
And notice, all I did is say, okay,

00:02:06.232 --> 00:02:07.175
F of seven.

00:02:07.175 --> 00:02:09.484
Well, if this is F of X--

00:02:09.484 --> 00:02:10.766
Let me write this down.

00:02:10.766 --> 00:02:12.933
So, if this is F of seven,

00:02:14.541 --> 00:02:16.450
if this is F of X,

00:02:16.450 --> 00:02:18.346
I just replaced the X with a seven

00:02:18.346 --> 00:02:19.786
to find F of seven,

00:02:19.786 --> 00:02:21.356
and we know that F of seven is also going

00:02:21.356 --> 00:02:22.418
to be equal to that.

00:02:22.418 --> 00:02:24.037
They gave us that information,

00:02:24.037 --> 00:02:26.617
but when you just look at this,

00:02:26.617 --> 00:02:28.866
it's pretty easy to figure
out what C is going to be.

00:02:28.866 --> 00:02:30.418
You can subtract five, E to the seven

00:02:30.418 --> 00:02:32.130
from both sides,

00:02:32.130 --> 00:02:34.963
and you see that C is equal to 40.

00:02:35.963 --> 00:02:38.774
And so we can rewrite F of X.

00:02:38.774 --> 00:02:41.592
We can say that F of X is equal

00:02:41.592 --> 00:02:43.259
to five, E to the X,

00:02:44.316 --> 00:02:45.983
plus C, which is 40.

00:02:47.264 --> 00:02:48.271
And so now, from that,

00:02:48.271 --> 00:02:50.281
we can evaluate F of zero.

00:02:50.281 --> 00:02:53.095
F of zero is going to be five times

00:02:53.095 --> 00:02:55.512
E to the zero power, plus 40.

00:02:56.884 --> 00:02:58.393
E to the zero is one,

00:02:58.393 --> 00:02:59.953
so it's gonna be five times one,

00:02:59.953 --> 00:03:02.042
which is just five, plus 40,

00:03:02.042 --> 00:03:03.792
which is equal to 45,

00:03:04.977 --> 00:03:06.310
and we're done.


Definite integral properties (no graph): breaking interval

WEBVTT
Kind: captions
Language: en

00:00:00.427 --> 00:00:01.846
- [Voiceover] We're
given the definitive role

00:00:01.846 --> 00:00:04.615
from one to four of f
of x dx is equal to six.

00:00:04.615 --> 00:00:06.209
And the definitive role for one to seven

00:00:06.209 --> 00:00:09.360
of f of x dx is equal to 11.

00:00:09.360 --> 00:00:11.335
And we wanna figure out
the definite integral

00:00:11.335 --> 00:00:14.252
from four to seven of f of x dx.

00:00:15.556 --> 00:00:18.518
So at least in my brain,
I'm visualizing these

00:00:18.518 --> 00:00:22.147
as areas between the
curve y equals f of x,

00:00:22.147 --> 00:00:24.105
and the x axis.

00:00:24.105 --> 00:00:26.114
So let's just draw that,
we don't know exactly

00:00:26.114 --> 00:00:28.730
what f of x is, but we can
draw an arbitrary f of x.

00:00:28.730 --> 00:00:30.955
Just to help us visualize things.

00:00:30.955 --> 00:00:35.727
So let me draw, so that is,
draw it in a bolder color.

00:00:35.727 --> 00:00:37.597
So that's our y axis.

00:00:37.597 --> 00:00:39.764
And this is our x axis.

00:00:41.589 --> 00:00:43.231
And let's see, all the action is happening

00:00:43.231 --> 00:00:46.312
between x, from x, from one to seven.

00:00:46.312 --> 00:00:48.979
So we can go one, we can go one,

00:00:50.649 --> 00:00:52.066
two, three, four,

00:00:54.195 --> 00:00:55.612
five, six, seven,

00:00:56.809 --> 00:00:58.404
and we can even go to eight if we like.

00:00:58.404 --> 00:01:00.090
But the important numbers,
let's see we're dealing

00:01:00.090 --> 00:01:02.840
with one, two, three, four, five,

00:01:03.732 --> 00:01:06.784
six, seven, and then we go to eight.

00:01:06.784 --> 00:01:10.336
And let me just draw the
graph y equals f of x,

00:01:10.336 --> 00:01:12.245
and I'm just gonna draw
something arbitrary here.

00:01:12.245 --> 00:01:15.853
So let's say the graph of y
equals f of x looks like that.

00:01:15.853 --> 00:01:17.603
Y is equal to f of x.

00:01:18.941 --> 00:01:21.174
And of course I put it on my axis.

00:01:21.174 --> 00:01:23.810
That's the x axis, that is the y axis.

00:01:23.810 --> 00:01:25.502
And let's think about what each of

00:01:25.502 --> 00:01:27.465
these integrals represent.

00:01:27.465 --> 00:01:31.108
So the integral, the definite
integral from one to four,

00:01:31.108 --> 00:01:33.585
well that's going to be,
we're going to be going

00:01:33.585 --> 00:01:36.418
from one to four, right over here.

00:01:38.868 --> 00:01:42.678
So this is the definite
integral from one to four.

00:01:42.678 --> 00:01:45.659
This area under the
curve, between the curve

00:01:45.659 --> 00:01:49.326
of the x axis, dx, which is equal to six.

00:01:50.503 --> 00:01:53.867
And now let's see, we also have the region

00:01:53.867 --> 00:01:56.284
that goes from four to seven.

00:01:57.964 --> 00:02:00.020
We have this region right over here.

00:02:00.020 --> 00:02:04.023
And that area is represented
by this definite integral,

00:02:04.023 --> 00:02:05.572
the one that we need to figure out.

00:02:05.572 --> 00:02:08.023
The definitive role from four to seven

00:02:08.023 --> 00:02:11.786
of f of x dx, we need to figure that out.

00:02:11.786 --> 00:02:13.673
And they also, what else do we have?

00:02:13.673 --> 00:02:15.406
So let me underline this.

00:02:15.406 --> 00:02:18.701
So that's that, this is
the area of the region

00:02:18.701 --> 00:02:20.543
between x equals four and x equals seven

00:02:20.543 --> 00:02:23.299
under y equals f of x above the x axis.

00:02:23.299 --> 00:02:26.999
And then they also gave us
this last piece of information.

00:02:26.999 --> 00:02:30.041
Which is, let me do this in another color,

00:02:30.041 --> 00:02:32.821
the definitive role from one to seven.

00:02:32.821 --> 00:02:34.710
Well that's going from one.

00:02:34.710 --> 00:02:36.850
One all the way to seven.

00:02:36.850 --> 00:02:40.889
So that's the sum of these
two regions right over here.

00:02:40.889 --> 00:02:44.188
So we could re-write this as the definite

00:02:44.188 --> 00:02:47.605
integral from one to four of f of x dx

00:02:48.592 --> 00:02:51.592
plus the definite integral from four

00:02:54.525 --> 00:02:57.608
to seven of f of x dx is equal to,

00:03:00.374 --> 00:03:03.707
the definite integral from one to seven.

00:03:04.885 --> 00:03:07.302
One to seven of f of x dx.

00:03:09.384 --> 00:03:11.026
And notice what's going on here.

00:03:11.026 --> 00:03:13.652
The first one just goes to the area from

00:03:13.652 --> 00:03:15.700
one to four, then we
go from four to seven,

00:03:15.700 --> 00:03:17.733
so if you add those together,
that's gonna be the area

00:03:17.733 --> 00:03:19.983
from one to seven, from one to seven.

00:03:19.983 --> 00:03:22.608
And so they gave us a
lot of this information,

00:03:22.608 --> 00:03:26.520
they tell us that this
right over here is six,

00:03:26.520 --> 00:03:28.087
let me do that same color.

00:03:28.087 --> 00:03:31.588
They tell us that this
is six, they tell us that

00:03:31.588 --> 00:03:33.755
this is 11, and so we have

00:03:35.942 --> 00:03:38.548
six plus this is equal to 11.

00:03:38.548 --> 00:03:40.643
Well six plus one is equal to 11.

00:03:40.643 --> 00:03:42.472
Well this thing right over here must be

00:03:42.472 --> 00:03:46.457
equal to this thing right,
the definite integral

00:03:46.457 --> 00:03:49.215
from four to seven must be equal to five.

00:03:49.215 --> 00:03:51.097
This must be equal to five.

00:03:51.097 --> 00:03:54.549
Another way to think about
it, if this region right

00:03:54.549 --> 00:03:57.062
over, having trouble switching colors,

00:03:57.062 --> 00:04:00.682
if this region right over here is six,

00:04:00.682 --> 00:04:04.682
so that has an area of
six, and the whole region

00:04:05.908 --> 00:04:08.575
if everything has an area of 11,

00:04:10.885 --> 00:04:14.209
so if that plus that has an area of 11,

00:04:14.209 --> 00:04:17.120
well then the stuff that we don't know,

00:04:17.120 --> 00:04:19.953
this orange region, is going to be

00:04:23.984 --> 00:04:25.067
11 minus six.

00:04:27.227 --> 00:04:29.007
So this region right over here is going to

00:04:29.007 --> 00:04:30.757
have an area of five.


Definite integral of trig function

WEBVTT
Kind: captions
Language: en

00:00:00.633 --> 00:00:01.831
- [Voiceover] Let's see if we can evaluate

00:00:01.831 --> 00:00:04.178
the definite integral from 11 pi over

00:00:04.178 --> 00:00:05.849
two to six pi

00:00:05.849 --> 00:00:07.849
of nine sine of x dx.

00:00:10.211 --> 00:00:11.663
So the first thing,
let's see if we can take

00:00:11.663 --> 00:00:13.807
the antiderivative of nine sine of x,

00:00:13.807 --> 00:00:15.113
and we could use some of our integration

00:00:15.113 --> 00:00:17.567
properties to simplify this a little bit.

00:00:17.567 --> 00:00:20.127
So this is going to be equal to,

00:00:20.127 --> 00:00:23.663
this is the same thing as
nine times the integral

00:00:23.663 --> 00:00:27.746
from 11 pi over two to
six pi of sine of x dx.

00:00:32.797 --> 00:00:35.133
And what's the
anitderivative of sine of x?

00:00:35.133 --> 00:00:38.741
Well we know, from our
derivatives, that the

00:00:38.741 --> 00:00:42.574
derivative with respect
to x of cosine of x

00:00:43.632 --> 00:00:46.215
is equal to negative sine of x,

00:00:47.435 --> 00:00:49.018
negative sine of x.

00:00:50.091 --> 00:00:51.611
So can we construct this in some way

00:00:51.611 --> 00:00:53.403
so this is a negative sine of x?

00:00:53.403 --> 00:00:55.107
What if I multiplied, on the inside,

00:00:55.107 --> 00:00:57.235
what if I multiplied it by negative one?

00:00:57.235 --> 00:00:58.820
Well I can't just
multiply in only one place

00:00:58.820 --> 00:00:59.981
by negative one, I need to multiply by

00:00:59.981 --> 00:01:02.561
negative one twice so I'm
not changing its value.

00:01:02.561 --> 00:01:05.755
So what if I said negative
nine times negative sine of x?

00:01:05.755 --> 00:01:07.567
Well this is still
gonna be nine sine of x.

00:01:07.567 --> 00:01:09.701
If you took negative nine
times negative sine of x,

00:01:09.701 --> 00:01:12.901
it is nine sine of x,
and I did it this way

00:01:12.901 --> 00:01:14.949
because now negative sine of x,

00:01:14.949 --> 00:01:18.629
it matches the derivative of cosine of x.

00:01:18.629 --> 00:01:21.865
So we could say that this
is all going to be equal to,

00:01:21.865 --> 00:01:23.457
it's all going to be
equal to, you have your

00:01:23.457 --> 00:01:28.038
negative nine out front,
negative nine times,

00:01:28.038 --> 00:01:30.621
times, I'll put it in brackets,

00:01:33.572 --> 00:01:36.366
negative nine times, the antiderivative

00:01:36.366 --> 00:01:39.000
of negative sine of x, well that

00:01:39.000 --> 00:01:41.892
is just going to be cosine of x,

00:01:41.892 --> 00:01:42.892
cosine of x,

00:01:44.484 --> 00:01:47.281
and we're going to
evaluate it at its bounds.

00:01:47.281 --> 00:01:50.748
We're going to evaluate it at six pi,

00:01:50.748 --> 00:01:53.356
let me do that in a
color I haven't used yet,

00:01:53.356 --> 00:01:55.507
we're gonna do that at six pi, and we're

00:01:55.507 --> 00:01:58.635
also going to do that at 11 pi over two,

00:01:58.635 --> 00:02:02.635
11 pi over two, so this
is going to be equal to,

00:02:03.671 --> 00:02:06.754
this is equal to negative nine times,

00:02:07.782 --> 00:02:09.949
I'm going to create some space here,

00:02:09.949 --> 00:02:12.323
so actually that's probably
more space than I need,

00:02:12.323 --> 00:02:15.627
it's going to be cosine of six pi,

00:02:15.627 --> 00:02:17.044
cosine of six pi,

00:02:23.306 --> 00:02:25.889
minus cosine of 11 pi over two,

00:02:26.989 --> 00:02:29.072
cosine of 11 pi over two.

00:02:35.369 --> 00:02:38.309
Well what is cosine of six pi going to be?

00:02:38.309 --> 00:02:40.645
Well, cosine of any multiple of

00:02:40.645 --> 00:02:43.624
two pi is going to be equal to one.

00:02:43.624 --> 00:02:45.195
You could view six pi as, we're going

00:02:45.195 --> 00:02:47.715
around the unit circle three times.

00:02:47.715 --> 00:02:50.279
So this is the same thing
as cosine of two pi,

00:02:50.279 --> 00:02:52.236
or the same thing as cosine of zero,

00:02:52.236 --> 00:02:54.438
so that is going to be equal to one.

00:02:54.438 --> 00:02:55.811
If that seems unfamiliar to you

00:02:55.811 --> 00:02:56.919
I encourage you to review the

00:02:56.919 --> 00:02:59.762
unit circle definition of cosine.

00:02:59.762 --> 00:03:02.845
And what is cosine of 11 pi over two?

00:03:04.496 --> 00:03:06.129
Let's see, let's subtract some,

00:03:06.129 --> 00:03:08.693
let's subtract some
multiple of two pi here

00:03:08.693 --> 00:03:11.930
to put it in values that
we can understand better.

00:03:11.930 --> 00:03:14.620
So this is, so let me write it here,

00:03:14.620 --> 00:03:18.787
cosine of 11 pi over 2,
that is the same thing as,

00:03:21.467 --> 00:03:24.509
let's see, if we were to subtract,

00:03:24.509 --> 00:03:27.288
this is the same thing as cosine of

00:03:27.288 --> 00:03:30.955
11 pi over two minus,
this is the same thing

00:03:32.416 --> 00:03:34.999
as five and one half pi, right?

00:03:36.000 --> 00:03:39.168
Yeah, so this is, so
we could view this as,

00:03:39.168 --> 00:03:42.677
we could subtract, let's subtract four pi,

00:03:42.677 --> 00:03:44.966
which is going to be, we could

00:03:44.966 --> 00:03:47.803
write that as eight pi over two.

00:03:47.803 --> 00:03:48.832
In fact, no, let's subtract five,

00:03:48.832 --> 00:03:52.999
no, let's subtract four pi,
which is eight pi over 2.

00:03:55.577 --> 00:03:56.951
So once again, I'm just subtracting

00:03:56.951 --> 00:03:59.032
a multiple of two pi,
which isn't gonna change

00:03:59.032 --> 00:04:01.996
the value of cosine, and
so this is going to be

00:04:01.996 --> 00:04:04.579
equal to cosine of 3 pi over 2.

00:04:07.036 --> 00:04:09.529
And if we imagine the unit circle,

00:04:09.529 --> 00:04:12.437
and let me draw the unit circle here.

00:04:12.437 --> 00:04:15.854
So it's my y axis, my x axis, and then

00:04:18.056 --> 00:04:21.594
I have the unit circle, so, whoops.

00:04:21.594 --> 00:04:23.692
So the unit circle just like that.

00:04:23.692 --> 00:04:26.189
So if we start at, this is zero,

00:04:26.189 --> 00:04:27.736
then you go to pi over two, then you

00:04:27.736 --> 00:04:30.657
go to pi, then you go
to three pi over two,

00:04:30.657 --> 00:04:33.062
so that's this point on the unit circle,

00:04:33.062 --> 00:04:34.630
so the cosine is the x coordinate,

00:04:34.630 --> 00:04:36.298
so this is going to be zero.

00:04:36.298 --> 00:04:39.658
This is zero so this
is zero, and so we get

00:04:39.658 --> 00:04:42.614
one minus zero, so
everything in the brackets

00:04:42.614 --> 00:04:46.211
evaluates out to one,
and so we are left with,

00:04:46.211 --> 00:04:50.449
so let me do that, so all
of this is equal to one.

00:04:50.449 --> 00:04:52.376
And so you have negative nine times one,

00:04:52.376 --> 00:04:55.914
which of course is just negative nine,

00:04:55.914 --> 00:04:58.860
is what this definite
integral evaluates to.


Definite integral properties (no graph): function combination

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.116
- [Voiceover] Given that
the definite integral

00:00:02.116 --> 00:00:03.987
from negative one to three of f of x dx

00:00:03.987 --> 00:00:05.641
is equal to negative two

00:00:05.641 --> 00:00:07.863
and the definite integral
from negative on to three

00:00:07.863 --> 00:00:10.876
of g of x dx is equal to five,

00:00:10.876 --> 00:00:14.052
what is the definite integral
from negative one to three

00:00:14.052 --> 00:00:17.552
of three f of x minus two g of x dx?

00:00:19.103 --> 00:00:21.066
All right, so to think
about this what we could use

00:00:21.066 --> 00:00:23.341
is some of our integration properties.

00:00:23.341 --> 00:00:26.297
And so the first thing
that I would want to do

00:00:26.297 --> 00:00:29.121
is we could split this
up into two integrals.

00:00:29.121 --> 00:00:31.048
We know that the

00:00:31.048 --> 00:00:34.145
and this is true of definite
or indefinite integrals,

00:00:34.145 --> 00:00:36.395
that the integral of f of x

00:00:38.172 --> 00:00:42.339
the integral of f of x
plus or minus g of x dx

00:00:44.556 --> 00:00:48.723
is going to be equal to
the integral of f of x dx

00:00:50.096 --> 00:00:53.679
plus or minus the integral of g of x dx.

00:00:54.553 --> 00:00:56.321
If this is a plus, this
is gonna be a plus.

00:00:56.321 --> 00:00:58.738
If this is a minus, this
is going to be a minus.

00:00:58.738 --> 00:01:01.034
So we could split this up in the same way.

00:01:01.034 --> 00:01:03.731
So this is going to be equal
to the definite integral

00:01:03.731 --> 00:01:07.731
from negative one to
three of three f of x dx

00:01:10.199 --> 00:01:11.784
minus the integral

00:01:11.784 --> 00:01:15.701
from negative one to
three of two g of x dx.

00:01:17.166 --> 00:01:19.543
Notice, all I did is I split it up.

00:01:19.543 --> 00:01:22.541
Taking the integral of the
difference of these functions

00:01:22.541 --> 00:01:24.200
is the same thing as taking

00:01:24.200 --> 00:01:27.446
the difference of the
integrals of those functions.

00:01:27.446 --> 00:01:29.812
Now the next thing we can do
is we can take the scalars,

00:01:29.812 --> 00:01:31.706
we're multiplying the
functions on the inside

00:01:31.706 --> 00:01:34.040
by these numbers, three and two,

00:01:34.040 --> 00:01:36.195
and we can take those
outside of the integral.

00:01:36.195 --> 00:01:37.615
And that comes straight
out of the property

00:01:37.615 --> 00:01:40.975
that if I'm taking the
integral of some constant

00:01:40.975 --> 00:01:44.591
times f of x dx, that
is equal to the constant

00:01:44.591 --> 00:01:47.508
times the integral of f of x dx,

00:01:48.694 --> 00:01:51.111
and so I can rewrite this as,

00:01:52.327 --> 00:01:54.460
so, let's see, I could
rewrite this first integral

00:01:54.460 --> 00:01:57.156
as three times the definite integral

00:01:57.156 --> 00:02:00.656
from negative one to three of f of x dx

00:02:06.035 --> 00:02:09.084
minus two times the definite integral

00:02:09.084 --> 00:02:12.167
from negative one to three of g of x,

00:02:13.304 --> 00:02:16.252
actually let me do the second
one in a different color.

00:02:16.252 --> 00:02:19.752
Minus, minus, this is gonna be in magenta.

00:02:20.790 --> 00:02:22.772
Minus two times the integral

00:02:22.772 --> 00:02:26.355
from negative one to three of g of x dx.

00:02:28.191 --> 00:02:30.482
And so what is this going to be equal to?

00:02:30.482 --> 00:02:33.612
Well they tell us, they tell
us what this thing is here

00:02:33.612 --> 00:02:35.990
that I'm underlining in orange,

00:02:35.990 --> 00:02:39.640
the integral from negative
one to three of f of x dx,

00:02:39.640 --> 00:02:42.519
they tell us that that
is equal to negative two.

00:02:42.519 --> 00:02:46.186
So that thing is negative
two, and likewise,

00:02:47.370 --> 00:02:49.802
this thing right over
here, the definite integral

00:02:49.802 --> 00:02:52.040
from negative one to three of g of x dx,

00:02:52.040 --> 00:02:54.965
they gave it right over
here, it's equal to five.

00:02:54.965 --> 00:02:56.635
So that's equal to five,

00:02:56.635 --> 00:02:59.376
and so the whole thing is going to be

00:02:59.376 --> 00:03:03.543
three times negative two,
which is equal to negative six

00:03:05.169 --> 00:03:08.322
minus two times five minus 10,

00:03:08.322 --> 00:03:10.822
which is equal to negative 16.

00:03:11.684 --> 00:03:12.934
And we're done!


Functions defined by integrals: switched interval

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.265
- [Voiceover] The graph
of f is shown below.

00:00:03.265 --> 00:00:07.018
Let g(x) be equal to the def intergral

00:00:07.018 --> 00:00:11.133
from zero to x, of f(t)dt.

00:00:11.133 --> 00:00:12.715
Now at first when you
see this you're like,

00:00:12.715 --> 00:00:14.588
wow this is strange, I have a function

00:00:14.588 --> 00:00:17.116
that is being defined by an integral,

00:00:17.116 --> 00:00:19.941
a def integral, but one
of it's bounds are X.

00:00:19.941 --> 00:00:21.857
You should just say well this is okay,

00:00:21.857 --> 00:00:23.955
a function can be defined any which way,

00:00:23.955 --> 00:00:25.878
and as we'll see, it's
actually quite straight forward

00:00:25.878 --> 00:00:27.195
to evaluate this.

00:00:27.195 --> 00:00:28.945
So g of negative two,

00:00:29.959 --> 00:00:31.459
g of negative two,

00:00:32.830 --> 00:00:34.720
and I'll do the negative
two in a different color,

00:00:34.720 --> 00:00:36.137
g of negative two,

00:00:36.137 --> 00:00:38.416
what we do is take this
expression over here,

00:00:38.416 --> 00:00:40.406
this def integral, and
where ever we see an X,

00:00:40.406 --> 00:00:41.968
we replace it with a negative two.

00:00:41.968 --> 00:00:44.612
So this is going to be equal to,

00:00:44.612 --> 00:00:46.699
the integral from zero to X,

00:00:46.699 --> 00:00:51.068
and I'll write X in a second, f(t)dt.

00:00:51.068 --> 00:00:55.535
Well X is now negative two,
this is now negative two.

00:00:55.535 --> 00:00:58.020
So how do we figure out what this is?

00:00:58.020 --> 00:00:59.991
Before we even look at this graph,

00:00:59.991 --> 00:01:02.501
you might say okay this
is the region under,

00:01:02.501 --> 00:01:04.731
the area of the region under the graph,

00:01:04.731 --> 00:01:08.762
y equals f(t), between
negative two and zero.

00:01:08.762 --> 00:01:11.103
But you have to be careful, notice,

00:01:11.103 --> 00:01:13.452
our upper bound here is
actually a lower number

00:01:13.452 --> 00:01:15.782
than our lower bound right over here.

00:01:15.782 --> 00:01:18.871
So it will be nice to swap those bounds

00:01:18.871 --> 00:01:21.900
so we can truly view it
as the area of the region

00:01:21.900 --> 00:01:26.067
under f(t), above the t axis,
between those two bounds.

00:01:27.163 --> 00:01:31.575
When you swap the bounds,
this is going to be equal to

00:01:31.575 --> 00:01:33.444
negative def integral,

00:01:33.444 --> 00:01:34.944
from negative two,

00:01:36.158 --> 00:01:37.908
negative two to zero,

00:01:39.645 --> 00:01:41.895
of f(t)dt,

00:01:42.801 --> 00:01:45.366
and now what we have right over here,

00:01:45.366 --> 00:01:47.325
what I'm squaring off in magenta,

00:01:47.325 --> 00:01:50.080
this is the area under the curve f,

00:01:50.080 --> 00:01:52.224
between negative two and zero.

00:01:52.224 --> 00:01:54.974
So between negative two and zero,

00:01:57.086 --> 00:01:58.897
so that is this area,

00:01:58.897 --> 00:02:02.206
right over here, that we care about.

00:02:02.206 --> 00:02:04.623
Now what is that going to be?

00:02:05.780 --> 00:02:07.368
There's a bunch of different
ways that you could do this,

00:02:07.368 --> 00:02:09.737
you could split it off
into a square and triangle.

00:02:09.737 --> 00:02:12.458
The area of this square here is four,

00:02:12.458 --> 00:02:16.634
it's two by two, make
sure to look at the units,

00:02:16.634 --> 00:02:19.117
sometimes each square doesn't
represent one square unit,

00:02:19.117 --> 00:02:21.385
in this case it does, so that's four,

00:02:21.385 --> 00:02:24.184
and then up here, this is half of four.

00:02:24.184 --> 00:02:26.380
If it was all of this, that would be four,

00:02:26.380 --> 00:02:30.519
this triangle is half
of four, so this is two.

00:02:30.519 --> 00:02:33.746
Or you could view this as base
times height times one half,

00:02:33.746 --> 00:02:36.382
which is going to be two
times two times one half.

00:02:36.382 --> 00:02:40.476
So this area is six, so this part is six,

00:02:40.476 --> 00:02:42.616
but we can't forget that negative sign,

00:02:42.616 --> 00:02:46.366
so this is going to be
equal to negative six.

00:02:48.151 --> 00:02:51.651
So g(-2) is negative six.