Line integral example 2 (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.740
In the last video, we set out
to figure out the surface area

00:00:03.740 --> 00:00:06.410
of the walls of this
weird-looking building, where

00:00:06.410 --> 00:00:10.880
the ceiling of the walls was
defined by the function f of xy

00:00:10.880 --> 00:00:15.800
is equal to x plus y squared,
and then the base of this

00:00:15.800 --> 00:00:20.030
building, or the contour of its
walls, was defined by the path

00:00:20.030 --> 00:00:24.100
where we have a circle of
radius 2 along here, then we go

00:00:24.100 --> 00:00:27.230
down along the y-axis, and then
we take another left, and we go

00:00:27.230 --> 00:00:29.450
along the x-axis, and
that was our building.

00:00:29.450 --> 00:00:31.790
And in the last video, we
figured out this first

00:00:31.790 --> 00:00:32.830
wall's surface area.

00:00:32.830 --> 00:00:36.570
In fact, you can think of it,
our original problem is, we

00:00:36.570 --> 00:00:40.730
wanted to figure out the line
integral along the closed path,

00:00:40.730 --> 00:00:44.360
so it was a closed line
integral, along the closed path

00:00:44.360 --> 00:00:51.770
c of f of xy, and we're always
multiplying f of xy times a

00:00:51.770 --> 00:00:54.720
little bit, a little, small
distance of our path, ds.

00:00:54.720 --> 00:00:57.550
We're writing this in the
most abstract way possible.

00:00:57.550 --> 00:00:59.800
And what we saw in the last
video is, the easiest way to do

00:00:59.800 --> 00:01:02.560
this is to break this up into
multiple paths, or into

00:01:02.560 --> 00:01:03.990
multiple problems.

00:01:03.990 --> 00:01:06.700
So you can imagine, this whole
contour, this whole path we

00:01:06.700 --> 00:01:08.900
call c, but we could call this
part, we figured out in

00:01:08.900 --> 00:01:11.330
the last video, c1.

00:01:11.330 --> 00:01:15.430
This part we can call, let me
make a pointer, c2, and this

00:01:15.430 --> 00:01:17.010
point right here is c3.

00:01:17.010 --> 00:01:21.640
So we could redefine, or we can
break up, this line integral,

00:01:21.640 --> 00:01:24.540
this closed-line integral, into
3 non-closed line integrals.

00:01:24.540 --> 00:01:30.340
This will be equal to the line
integral along the path c1 of f

00:01:30.340 --> 00:01:40.560
of xy ds, plus the line
integral along c2 of f of x y

00:01:40.560 --> 00:01:44.900
ds plus the line integral, you
might have guessed it, along c3

00:01:44.900 --> 00:01:50.930
of f of xy ds, and in the last
video, we got as far as

00:01:50.930 --> 00:01:53.340
figuring out this first part,
this first curvy wall

00:01:53.340 --> 00:01:54.490
all right here.

00:01:54.490 --> 00:01:59.870
Its surface area, we figured
out, was 4 plus 2 pi.

00:01:59.870 --> 00:02:02.960
Now we've got to figure
out the other 2 parts.

00:02:02.960 --> 00:02:06.850
So let's do C2, let's do
this line integral next.

00:02:06.850 --> 00:02:08.600
And in order to do it,
we need to do another

00:02:08.600 --> 00:02:10.540
parameterization of x and y.

00:02:10.540 --> 00:02:12.620
It's going to be different than
what we did for this part.

00:02:12.620 --> 00:02:14.570
We're no longer along
this circle, we're

00:02:14.570 --> 00:02:16.390
just along the y-axis.

00:02:16.390 --> 00:02:18.570
So as long as we're there,
x is definitely going

00:02:18.570 --> 00:02:19.750
to be equal to 0.

00:02:19.750 --> 00:02:22.060
So that's my parameterization,
x is equal to 0.

00:02:22.060 --> 00:02:25.070
If we're along the y-axis, x
is definitely equal to 0.

00:02:25.070 --> 00:02:30.710
And then y, we could say it
starts off at y is equal to 2.

00:02:30.710 --> 00:02:37.790
Maybe we'll say y is equal to 2
minus t, for t is between 0, t

00:02:37.790 --> 00:02:40.230
is greater than or equal to
0, less than or equal to 2.

00:02:40.230 --> 00:02:41.270
And that should work.

00:02:41.270 --> 00:02:44.300
When t is equal to 0, we're at
this point right there, and

00:02:44.300 --> 00:02:47.900
then as t increases towards 2,
we move down the y-axis, until

00:02:47.900 --> 00:02:50.270
eventually when t is equal
to 2, we're at that

00:02:50.270 --> 00:02:51.080
point right there.

00:02:51.080 --> 00:02:52.780
So that's our parameterization.

00:02:52.780 --> 00:02:55.530
And so let's evaluate this
line, and we could do our

00:02:55.530 --> 00:02:57.250
derivatives, too, if we like.

00:02:57.250 --> 00:02:59.920
What's the derivative,
I'll write it over here.

00:02:59.920 --> 00:03:01.290
What's dx dt?

00:03:01.290 --> 00:03:02.230
Pretty straightforward.

00:03:02.230 --> 00:03:09.550
Derivative of 0 is 0, and
dy dt is equal to the

00:03:09.550 --> 00:03:10.400
derivative of this.

00:03:10.400 --> 00:03:11.930
It's just minus 1, right?

00:03:11.930 --> 00:03:17.030
2 minus t, derivative of
minus t, is just minus 1.

00:03:17.030 --> 00:03:19.940
And so let's just break it up.

00:03:19.940 --> 00:03:23.540
So we have this thing right
here, so we have the

00:03:23.540 --> 00:03:25.635
integral along c2.

00:03:25.635 --> 00:03:28.490
But let's, instead of writing
c2, I'll leave c2 there, but

00:03:28.490 --> 00:03:35.450
we'll say were going from t is
equal to 0 to 2 of f of xy.

00:03:35.450 --> 00:03:42.690
f of xy is this thing right
here, is x plus y squared,

00:03:42.690 --> 00:03:44.090
and then times ds.

00:03:47.420 --> 00:03:50.410
And we know from the last
several videos, ds can be

00:03:50.410 --> 00:03:57.350
rewritten as the square root of
dx dt squared, so 0 squared,

00:03:57.350 --> 00:04:02.340
plus dy dt squared, so minus 1
squared is 1, all

00:04:02.340 --> 00:04:03.910
of that times dt.

00:04:03.910 --> 00:04:07.390
And obviously, this is
pretty nice and clean.

00:04:07.390 --> 00:04:11.640
This is 0 plus 1, square
root, this just becomes 1.

00:04:11.640 --> 00:04:13.260
And then what is x?

00:04:13.260 --> 00:04:16.330
x, if we write it in terms of
our parameterization, is always

00:04:16.330 --> 00:04:19.780
going to be equal to 0, and
then y squared is going to

00:04:19.780 --> 00:04:21.280
be 2 minus is t squared.

00:04:21.280 --> 00:04:24.490
So this is going to be
2 minus t squared.

00:04:24.490 --> 00:04:28.020
So this whole crazy thing
simplified to, we're going to

00:04:28.020 --> 00:04:34.240
go from t is equal to 0 to t is
equal to 2, the x disappears in

00:04:34.240 --> 00:04:37.330
our parameterization, x just
stays 0, regardless of what

00:04:37.330 --> 00:04:41.070
t is, and then you have y
squared, but y is the same

00:04:41.070 --> 00:04:47.180
thing as 2 minus t, so 2 minus
t squared, and then you have

00:04:47.180 --> 00:04:48.440
your dt sitting out there.

00:04:48.440 --> 00:04:49.840
This is pretty straightforward.

00:04:49.840 --> 00:04:52.910
I always find it easier when
you're finding an

00:04:52.910 --> 00:04:55.780
antiderivative of this,
although you can do this in

00:04:55.780 --> 00:04:59.610
your head, I like to
just actually multiply

00:04:59.610 --> 00:05:00.720
out this binomial.

00:05:00.720 --> 00:05:04.340
So this is going to be equal to
the antiderivative from t is

00:05:04.340 --> 00:05:16.230
equal to 0 to t is equal to 2
of 4 minus 2 minus 4t plus t

00:05:16.230 --> 00:05:21.690
squared, plus t squared,
just like that dt.

00:05:21.690 --> 00:05:23.290
And this is pretty
straightforward.

00:05:23.290 --> 00:05:27.150
This is going to be, the
antiderivative of this is 4

00:05:27.150 --> 00:05:32.750
t minus 2 t squared, right?

00:05:32.750 --> 00:05:34.580
When you take the derivative,
there's 2 times minus 2 is

00:05:34.580 --> 00:05:43.200
minus 4 t, and then you have
plus 1/3 t to the third, right?

00:05:43.200 --> 00:05:45.190
These are just simple
antiderivatives, and we need

00:05:45.190 --> 00:05:48.300
to evaluate it from 0 to 2.

00:05:48.300 --> 00:05:50.180
And so let's evaluate it at 2.

00:05:50.180 --> 00:05:54.140
4 times 2 is 8, let
me pick a new color.

00:05:54.140 --> 00:06:02.560
4 times 2 is 8, minus 2 times 2
squared, so 2 times 4, so minus

00:06:02.560 --> 00:06:08.800
8, plus 1/3 times 2
to the third power.

00:06:08.800 --> 00:06:10.910
So 1/3 times 8.

00:06:10.910 --> 00:06:11.970
So these cancel out.

00:06:11.970 --> 00:06:15.530
We have 8 minus 8, and
we just have 8/3.

00:06:15.530 --> 00:06:17.530
So this just becomes 8/3.

00:06:17.530 --> 00:06:20.470
And then we have to put a 0 in,
minus 0 evaluate here, but

00:06:20.470 --> 00:06:21.320
it's just going to be 0.

00:06:21.320 --> 00:06:24.180
We have 4 times 0, two times 0,
all of these are going to be 0.

00:06:24.180 --> 00:06:25.730
So minus 0.

00:06:25.730 --> 00:06:27.700
So just like that, we
found our surface area

00:06:27.700 --> 00:06:30.210
of our second wall.

00:06:30.210 --> 00:06:34.250
This turned out being,
this right here is 8/3.

00:06:34.250 --> 00:06:35.890
And now we have our last
wall, and then we can

00:06:35.890 --> 00:06:37.380
just add them up.

00:06:37.380 --> 00:06:40.640
So we have our last wall.

00:06:40.640 --> 00:06:41.800
I'll do another
parameterization.

00:06:41.800 --> 00:06:44.180
I want to have the graph there.

00:06:44.180 --> 00:06:46.620
Well, maybe I can
paste it again.

00:06:46.620 --> 00:06:46.990
Edit.

00:06:46.990 --> 00:06:49.590
So there's the graph again.

00:06:49.590 --> 00:06:52.250
And now we're going
to do our last wall.

00:06:52.250 --> 00:06:54.870
So our last wall is this one
right here, which is, we

00:06:54.870 --> 00:06:56.930
could write it, you
know, this was c3.

00:06:56.930 --> 00:06:59.750
Let me switch colors here.

00:06:59.750 --> 00:07:06.420
So this is c, we're going to go
along contour c3 of f of xy ds,

00:07:06.420 --> 00:07:10.170
which is the same thing as,
let's do a parameterization.

00:07:10.170 --> 00:07:14.580
Along this curve, if we just
say, x is equal to t, very

00:07:14.580 --> 00:07:17.770
straight forward, for t is
greater than or equal to 0,

00:07:17.770 --> 00:07:20.420
less than or equal to 2, and
this whole time that we're

00:07:20.420 --> 00:07:23.520
along the x-axis, y is
going to be equal to 0.

00:07:23.520 --> 00:07:25.650
That's pretty straightforward
parameterization.

00:07:25.650 --> 00:07:29.100
So this is going to be equal
to, we're going to go from t is

00:07:29.100 --> 00:07:37.845
equal to 0 to t is equal to 2
of f of xy, which is, I'll

00:07:37.845 --> 00:07:41.860
write in terms of x right now,
x and y, x plus y

00:07:41.860 --> 00:07:44.760
squared times ds.

00:07:44.760 --> 00:07:47.970
Now, what is dx-- well, let
me write ds right here.

00:07:47.970 --> 00:07:48.690
Times ds.

00:07:48.690 --> 00:07:50.520
That's what we're dealing with.

00:07:50.520 --> 00:07:52.870
Now we know what ds is.

00:07:52.870 --> 00:08:01.030
ds is equal to the square
root of dx dt squared plus

00:08:01.030 --> 00:08:04.940
dy dt squared times dt.

00:08:04.940 --> 00:08:06.510
We proved that in
the first video.

00:08:06.510 --> 00:08:08.690
Or we didn't rigorously prove
it, but we got the sense

00:08:08.690 --> 00:08:09.980
of why this is true.

00:08:09.980 --> 00:08:12.560
And what's the derivative
of x with respect to t?

00:08:12.560 --> 00:08:15.890
Well, that's just 1, so this
is just going to be a 1,

00:08:15.890 --> 00:08:17.250
1 squared, same thing.

00:08:17.250 --> 00:08:19.030
And the derivative of y
with respect to z is 0.

00:08:19.030 --> 00:08:22.840
So this is is 0, 1 plus 0 is
1, square root of 1 is 1.

00:08:22.840 --> 00:08:27.090
So this thing just becomes dt.

00:08:27.090 --> 00:08:30.290
ds is going to be equal
to dt, in this case.

00:08:30.290 --> 00:08:33.510
So this just becomes a dt.

00:08:33.510 --> 00:08:37.515
And then our x is going to be
equal to a t, that's part

00:08:37.515 --> 00:08:40.460
of our definition of our
parameterization, and y is

00:08:40.460 --> 00:08:42.090
zero, so we can ignore it.

00:08:42.090 --> 00:08:43.710
So this was a
super-simple integral.

00:08:43.710 --> 00:08:47.580
So this simplified down to,
we're going to go from 0 to 2

00:08:47.580 --> 00:08:54.540
of t dt, which is equal to the
antiderivative of t is just 1/2

00:08:54.540 --> 00:08:58.240
t squared, and we're going to
go 0 to 2, which is equal

00:08:58.240 --> 00:09:01.160
to 1/2 times 2 squared.

00:09:01.160 --> 00:09:05.780
2 squared is 4, times 1/2
is 2, and then minus 1/2

00:09:05.780 --> 00:09:08.410
times 0 squared, minus 0.

00:09:08.410 --> 00:09:12.860
So this third wall's area
right there is just 2.

00:09:12.860 --> 00:09:15.390
Pretty straightforward.

00:09:15.390 --> 00:09:19.580
So that right there, the
area there, is just 2.

00:09:19.580 --> 00:09:23.270
And so to answer our question,
what was this line integral

00:09:23.270 --> 00:09:27.420
evaluated over this
closed path of f of xy?

00:09:27.420 --> 00:09:28.870
Well, we just add
up these numbers.

00:09:28.870 --> 00:09:32.980
We have 4 plus 2 pi plus 8/3
plus 2, well, what is this.

00:09:32.980 --> 00:09:39.530
8/3 is same thing as 2 and 2/3,
so we have 4 plus 2 and 2/3 is

00:09:39.530 --> 00:09:44.290
6 and 2/3, plus another 2 is 8
and 2/3, so this whole thing

00:09:44.290 --> 00:09:47.880
becomes 8 and 2/3, if we
write it as a mixed

00:09:47.880 --> 00:09:50.470
number, plus 2 pi.

00:09:50.470 --> 00:09:51.550
And we're done!

00:09:51.550 --> 00:09:52.190
And we're done.

00:09:52.190 --> 00:09:55.210
Now we can start trying to
do line integrals with

00:09:55.210 --> 00:09:57.460
vector-valued functions.


Line integral example 2 (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:05.480
So let's say I have a function
of x and y; f of x and y is

00:00:05.480 --> 00:00:08.470
equal to x plus y squared.

00:00:08.470 --> 00:00:11.400
If I try to draw that,
let's see if I can have

00:00:11.400 --> 00:00:13.100
a good attempt at it.

00:00:13.100 --> 00:00:16.590
That is my y axis-- I'm going
to do a little perspective here

00:00:16.590 --> 00:00:21.440
--this is my x axis-- I make do
the negative x and y axis,

00:00:21.440 --> 00:00:24.180
could do it in that direction
--this is my x axis here.

00:00:24.180 --> 00:00:27.150
And if I were to graph this
when y is 0, it's going to be

00:00:27.150 --> 00:00:29.720
just a-- let me draw it in
yellow --is going to be just a

00:00:29.720 --> 00:00:32.310
straight line that looks
something like that.

00:00:32.310 --> 00:00:34.050
And then for any given
actually, we're going to

00:00:34.050 --> 00:00:36.010
have a parabola in y.

00:00:36.010 --> 00:00:39.230
y is going to look
something like that.

00:00:39.230 --> 00:00:40.335
I'm just going to it in
the positive quadrant.

00:00:40.335 --> 00:00:41.780
It's going to look
something like that.

00:00:44.980 --> 00:00:47.090
It'll actually, when you go
into the negative y, you're

00:00:47.090 --> 00:00:48.905
going to see the other half of
the parabola, but I'm not going

00:00:48.905 --> 00:00:50.370
to worry about it too much.

00:00:50.370 --> 00:00:51.980
So you're going to
have this surface.

00:00:51.980 --> 00:00:53.190
it looks something like that.

00:00:53.190 --> 00:00:55.140
Maybe I'll do to another
attempt at drawing it.

00:00:55.140 --> 00:00:57.930
But this is our ceiling we're
going to deal with again.

00:00:57.930 --> 00:01:03.340
And then I'm going to have
a path in the xy plane.

00:01:03.340 --> 00:01:08.660
I'm going to start at the
point 2 comma 0. x is

00:01:08.660 --> 00:01:10.430
equal to 2, y is 0.

00:01:10.430 --> 00:01:13.670
And I'm going to travel, just
like we did in the last video,

00:01:13.670 --> 00:01:16.380
I'm going to travel along a
circle, but this time the

00:01:16.380 --> 00:01:18.630
circle's going to
have of radius 2.

00:01:18.630 --> 00:01:20.650
Move counter clockwise
in that circle.

00:01:20.650 --> 00:01:22.610
This is on the xy plane,
just to be able to

00:01:22.610 --> 00:01:24.040
visualize it properly.

00:01:24.040 --> 00:01:26.500
So this right here's
a point 0, 2.

00:01:26.500 --> 00:01:30.055
And I'm going to come
back along the y axis.

00:01:30.055 --> 00:01:35.640
This is my path; I'm going to
come back along the y access

00:01:35.640 --> 00:01:38.220
and then so I look a left here,
and then I'm going to take

00:01:38.220 --> 00:01:40.640
another left here in and
come back along the x axis.

00:01:44.930 --> 00:01:46.540
I drew it in these
two shades of green.

00:01:46.540 --> 00:01:48.680
That is my contour.

00:01:48.680 --> 00:01:52.830
And what I want to do is I want
to evaluate the surface area

00:01:52.830 --> 00:01:56.260
of essentially this little
building that has the roof of f

00:01:56.260 --> 00:02:00.020
of xy is equal to x plus y
squared, and I want to find the

00:02:00.020 --> 00:02:01.620
surface area of its walls.

00:02:01.620 --> 00:02:05.705
So you'll have this wall right
here, whose base is the x axis.

00:02:05.705 --> 00:02:09.250
Then you're going to have this
wall, which is along the curve;

00:02:09.250 --> 00:02:14.490
it's going to look something
like kind of funky wall on

00:02:14.490 --> 00:02:16.410
that curved side right there.

00:02:16.410 --> 00:02:19.550
I'll try my best effort to
try to-- it's going to be

00:02:19.550 --> 00:02:25.510
curving way up like that
and then along the y axis.

00:02:28.270 --> 00:02:32.240
It's going to have like a half
a parabolic wall right there.

00:02:32.240 --> 00:02:34.150
I'll do that back wall
along the y axis.

00:02:34.150 --> 00:02:38.400
I'll do that in orange,
I'll use magenta.

00:02:38.400 --> 00:02:41.530
That is the back wall
along the y axis.

00:02:41.530 --> 00:02:45.660
Then you have this front
wall along the x axis.

00:02:45.660 --> 00:02:50.230
And then you have this weird
curvy curtain or wall-- do that

00:02:50.230 --> 00:02:55.020
maybe in blue --that goes along
this curve right here, this

00:02:55.020 --> 00:02:56.740
part of a circle of radius 2.

00:02:56.740 --> 00:02:58.890
So hopefully you get
that visualization.

00:02:58.890 --> 00:03:00.670
It's a little harder; I'm
not using any graphic

00:03:00.670 --> 00:03:01.680
program at this time.

00:03:01.680 --> 00:03:03.900
But I want to figure out
the surface area, the

00:03:03.900 --> 00:03:06.270
combined surface area
of these three walls.

00:03:06.270 --> 00:03:09.390
And in very simple notation we
could say, well, the surface

00:03:09.390 --> 00:03:16.430
area of those walls-- of this
wall plus that wall plus that

00:03:16.430 --> 00:03:22.570
wall --is going to be equal to
the line integral along this

00:03:22.570 --> 00:03:26.020
curve, or along this contour--
however you want to call it

00:03:26.020 --> 00:03:32.820
--of f of xy,-- so that's x
plus y squared --ds, where ds

00:03:32.820 --> 00:03:35.220
is just a little length
along our contour.

00:03:35.220 --> 00:03:37.450
And since this is a closed
loop, we'll call this a

00:03:37.450 --> 00:03:38.500
closed line interval.

00:03:38.500 --> 00:03:40.560
And we'll sometimes see
this notation right here.

00:03:43.080 --> 00:03:44.480
Often you'll see that
in physics books.

00:03:44.480 --> 00:03:45.870
And we'll be dealing
with a lot more.

00:03:45.870 --> 00:03:47.330
And we'll put a circle
on the interval sign.

00:03:47.330 --> 00:03:50.060
And all that means is that the
contour we're dealing with is a

00:03:50.060 --> 00:03:53.750
closed contour; we get back
to where we started from.

00:03:53.750 --> 00:03:55.770
But how do we solve this thing?

00:03:55.770 --> 00:03:57.660
A good place to start
is to just to find

00:03:57.660 --> 00:03:58.630
the contour itself.

00:03:58.630 --> 00:04:00.830
And just to simply it, we're
going to divide it into three

00:04:00.830 --> 00:04:03.040
pieces and it essentially just
do three separate

00:04:03.040 --> 00:04:04.290
line integrals.

00:04:04.290 --> 00:04:08.770
Because you know, this isn't
a very continuous contour.

00:04:08.770 --> 00:04:10.050
so the first part.

00:04:10.050 --> 00:04:11.980
Let's do this first part of
the curve where we're going

00:04:11.980 --> 00:04:15.640
along a circle of radius 2.

00:04:15.640 --> 00:04:20.770
And that's pretty easy to
construct if we have x-- let me

00:04:20.770 --> 00:04:25.850
do each part of the contour in
a different color, so if I do

00:04:25.850 --> 00:04:30.920
orange this part of the contour
--if we say that x is equal 2

00:04:30.920 --> 00:04:39.440
cosine of t and y is equal to 2
sine of t and if we say that

00:04:39.440 --> 00:04:42.680
t-- and this is really just
building off what we saw on the

00:04:42.680 --> 00:04:46.960
last video --if we say that t--
and that this is from t is a

00:04:46.960 --> 00:04:54.170
greater than or equal to 0 and
is less than or equal to pi

00:04:54.170 --> 00:04:58.270
over 2 --t is essentially going
to be the angle that

00:04:58.270 --> 00:05:00.200
we're going along the
circle right here.

00:05:00.200 --> 00:05:02.040
This will actually
describe this path.

00:05:02.040 --> 00:05:03.820
And if you know, how I
constructed this is little

00:05:03.820 --> 00:05:05.980
confusing, you might want
to review the video on

00:05:05.980 --> 00:05:07.860
parametric equations.

00:05:07.860 --> 00:05:09.360
So this is the first
part of our path.

00:05:09.360 --> 00:05:14.080
So if we just wanted to find
the surface area of that wall

00:05:14.080 --> 00:05:15.940
right there, we know we're
going to have to find

00:05:15.940 --> 00:05:17.670
dx, dt and dy, dt.

00:05:17.670 --> 00:05:19.970
So let's get that out
of the way right now.

00:05:19.970 --> 00:05:28.770
So if we say dx, dt is going to
be equal to minus 2, sine of t,

00:05:28.770 --> 00:05:34.495
dy, dy is going to be equal to
2 cosine of t; just the

00:05:34.495 --> 00:05:35.840
derivatives of these.

00:05:35.840 --> 00:05:37.350
We've seen that
many times before.

00:05:37.350 --> 00:05:41.290
So it we want this orange
wall's surface area, we can

00:05:41.290 --> 00:05:43.675
take the integral-- and if any
of this is confusing, there are

00:05:43.675 --> 00:05:47.400
two videos before this where we
kind of derive this formula

00:05:47.400 --> 00:05:51.170
--but we could take the
integral from t is equal to 0

00:05:51.170 --> 00:05:59.260
to pi over 2 our function of x
plus y squared and

00:05:59.260 --> 00:06:01.610
then times the ds.

00:06:01.610 --> 00:06:03.570
So x plus y squared will
give the height of

00:06:03.570 --> 00:06:04.550
each little block.

00:06:04.550 --> 00:06:06.360
And then we want to get the
width of each little block,

00:06:06.360 --> 00:06:10.690
which is ds, but we know that
we can rewrite the ds as the

00:06:10.690 --> 00:06:18.930
square root-- give myself some
room right here --of dx of the

00:06:18.930 --> 00:06:24.310
derivative of x with respect to
t squared-- so that is minus 2

00:06:24.310 --> 00:06:29.160
sine of t squared --plus
the derivative of y with

00:06:29.160 --> 00:06:36.660
respect to t squared, dt.

00:06:36.660 --> 00:06:39.050
This will give us the orange
section, and then we can worry

00:06:39.050 --> 00:06:41.820
about the other two walls.

00:06:41.820 --> 00:06:43.410
And so how can we
simplify this?

00:06:43.410 --> 00:06:49.670
Well, this is going to be equal
to the integral from 0 to pi

00:06:49.670 --> 00:06:55.000
over 2 of x plus y squared.

00:06:55.000 --> 00:06:59.180
And actually, let me write
everything in terms of t.

00:06:59.180 --> 00:07:01.250
So x is equal to 2 cosine of t.

00:07:01.250 --> 00:07:02.910
So let me write that down.

00:07:02.910 --> 00:07:13.310
So it's 2 cosine of t plus y,
which is 2 sine of t, and we're

00:07:13.310 --> 00:07:14.830
going to square everything.

00:07:14.830 --> 00:07:18.780
And then all of that times
this crazy radical.

00:07:18.780 --> 00:07:22.880
Right now it looks like a hard
antiderivative or integral to

00:07:22.880 --> 00:07:24.780
solve, but I we'll find
out it's not too bad.

00:07:24.780 --> 00:07:30.380
This is going to be equal to
4 sine squared of t plus

00:07:30.380 --> 00:07:34.380
4 cosine squared of t.

00:07:34.380 --> 00:07:37.020
We can factor a 4 out.

00:07:37.020 --> 00:07:39.450
I don't want to forget the dt.

00:07:39.450 --> 00:07:41.820
This over here-- let me just
simplify this expression so I

00:07:41.820 --> 00:07:43.260
don't have to keep
rewriting it.

00:07:43.260 --> 00:07:47.820
That is the same thing is
the square root of 4 times

00:07:47.820 --> 00:07:52.650
sine squared of t plus
cosine squared of t.

00:07:52.650 --> 00:07:54.800
We know what that
is: that's just 1.

00:07:54.800 --> 00:07:56.516
So this whole thing just
simplifies to the square

00:07:56.516 --> 00:07:58.650
root of 4, which is just 2.

00:07:58.650 --> 00:08:02.120
So this whole thing simplifies
to 2, which is nice for

00:08:02.120 --> 00:08:04.290
solving our antiderivative.

00:08:04.290 --> 00:08:05.860
That means simplifying
things a lot.

00:08:05.860 --> 00:08:09.620
So this whole thing simplifies
down to-- I'll do it over here.

00:08:09.620 --> 00:08:12.320
I don't want to waste too much
space; I have two more walls to

00:08:12.320 --> 00:08:16.760
figure out --the integral from
t is equal to 0 to pi over 2.

00:08:16.760 --> 00:08:17.660
I want to make it very clear.

00:08:17.660 --> 00:08:20.700
I just chose the simplest
parametrization I

00:08:20.700 --> 00:08:22.080
could for x and y.

00:08:22.080 --> 00:08:23.630
But I could have picked
other parametrizations, but

00:08:23.630 --> 00:08:25.250
then I would have had to
change t accordingly.

00:08:25.250 --> 00:08:28.200
So as long as you're consistent
with how you do it, it

00:08:28.200 --> 00:08:29.020
should all work out.

00:08:29.020 --> 00:08:32.290
There isn't just one
parametrization for this curve;

00:08:32.290 --> 00:08:34.130
it's kind of depending on how
fast you want to go

00:08:34.130 --> 00:08:35.140
along the curve.

00:08:35.140 --> 00:08:39.450
Watch the parametric functions
videos if you want a little

00:08:39.450 --> 00:08:40.900
bit more depth on that.

00:08:40.900 --> 00:08:41.860
Anyway, this thing simplifies.

00:08:41.860 --> 00:08:44.400
We have a 2 here; 2 times
cosine of t, that's

00:08:44.400 --> 00:08:47.720
4 cosine of t.

00:08:47.720 --> 00:08:52.500
And then here we have 2 sine
squared sine of t squared.

00:08:52.500 --> 00:08:54.430
So that's 4 sine squared of t.

00:08:58.390 --> 00:09:01.230
And then we have to multiply
times this 2 again, so

00:09:01.230 --> 00:09:02.970
that gives us an 8.

00:09:02.970 --> 00:09:07.750
8 time sine squared of t, dt.

00:09:07.750 --> 00:09:10.040
And then you know, sine squared
of t; that looks like a

00:09:10.040 --> 00:09:12.840
tough thing to find the
antiderivative for, but we can

00:09:12.840 --> 00:09:18.150
remember that sine squared of,
really anything-- we could say

00:09:18.150 --> 00:09:21.660
sine squared of u is equal
to 1/2 half times 1

00:09:21.660 --> 00:09:24.290
minus cosine of 2u.

00:09:24.290 --> 00:09:26.700
So we can reuse this identity.

00:09:26.700 --> 00:09:30.380
I can try the t here; sine
squared of t is equal to 1/2

00:09:30.380 --> 00:09:33.540
times 1 minus cosine of 2t.

00:09:33.540 --> 00:09:35.050
Let me rewrite it that way
because that'll make the

00:09:35.050 --> 00:09:36.390
integral a lot easier to solve.

00:09:36.390 --> 00:09:40.530
So we get integral from 0 the
pi over 2-- and actually I

00:09:40.530 --> 00:09:45.080
could break up, well I won't
break it up --of 4 cosine of

00:09:45.080 --> 00:09:51.550
t plus 8 times this thing.

00:09:51.550 --> 00:09:53.350
8 times this thing; this
is the same thing as

00:09:53.350 --> 00:09:55.020
sine squared of t.

00:09:55.020 --> 00:10:02.250
So 8 times this-- 8 times 1/2
is 4 --4 times 1 minus cosine

00:10:02.250 --> 00:10:05.590
of 2t-- just use a little trig
identity there --and

00:10:05.590 --> 00:10:07.390
all of that dt.

00:10:07.390 --> 00:10:09.430
Now this should be reasonably
straight forward to get

00:10:09.430 --> 00:10:10.800
the antiderivative of.

00:10:10.800 --> 00:10:11.980
Let's just take it.

00:10:11.980 --> 00:10:16.500
The antiderivative of this
is antiderivative of cosine

00:10:16.500 --> 00:10:18.610
of t; that's a sine of t.

00:10:18.610 --> 00:10:20.150
The derivative of
sine is cosine.

00:10:20.150 --> 00:10:25.050
So this is going to be 4 sine
of t-- the scalars don't affect

00:10:25.050 --> 00:10:28.470
anything --and then, well let
me just distribute this 4.

00:10:28.470 --> 00:10:34.640
So this is 4 times 1 which
is 4 minus 4 cosine of 2t.

00:10:34.640 --> 00:10:39.630
So the antiderivative of 4 is
4t-- plus 4t --and then the

00:10:39.630 --> 00:10:43.880
antiderivative of minus
4 cosine of u00b5 t?

00:10:43.880 --> 00:10:45.970
Let's see it's going
to be sine of 2t.

00:10:51.590 --> 00:10:58.900
The derivative of sine of
2t is 2 cosine of 2t.

00:10:58.900 --> 00:11:01.660
We're going to have to have a
minus sign there, and put a 2

00:11:01.660 --> 00:11:03.420
there, and now it
should work out.

00:11:03.420 --> 00:11:05.740
What's the derivative
of minus 2 sine of t?

00:11:05.740 --> 00:11:07.240
Take the derivative of
the inside 2 times

00:11:07.240 --> 00:11:09.400
minus 2 is minus 4.

00:11:09.400 --> 00:11:11.740
And the derivative of sine
of 2t with respect to

00:11:11.740 --> 00:11:13.280
2t is cosine of 2t.

00:11:13.280 --> 00:11:15.830
So there we go; we've figured
out our antiderivative.

00:11:15.830 --> 00:11:19.260
Now we evaluate it
from 0 the pi over 2.

00:11:22.000 --> 00:11:23.420
And what do we get?

00:11:23.420 --> 00:11:27.210
We get 4 sine-- let me write
this down, for I don't want to

00:11:27.210 --> 00:11:34.100
skip too many --sine of pi over
2 plus 4 times pi over 2--

00:11:34.100 --> 00:11:42.040
that's just 2 pi minus 2 sine
of 2 times pi over 2 sine of

00:11:42.040 --> 00:11:48.620
pie, and then all of that minus
all this evaluated at 0.

00:11:48.620 --> 00:11:50.580
That's actually pretty
straightforward because

00:11:50.580 --> 00:11:52.670
sine of 0 is 0.

00:11:52.670 --> 00:11:56.490
4 times 0 is 0, and sine of
2 times 0, that's also 0.

00:11:56.490 --> 00:11:59.190
So everything with the
0's work out nicely.

00:11:59.190 --> 00:12:00.350
And then what do we have here?

00:12:00.350 --> 00:12:05.770
Sine of pi over 2-- in my head,
I think sine of 90 degrees;

00:12:05.770 --> 00:12:09.320
same thing --that is 1.

00:12:09.320 --> 00:12:11.980
And then sine of pi is
0, that's 180 degrees.

00:12:11.980 --> 00:12:13.670
So this whole thing
cancels out.

00:12:13.670 --> 00:12:17.440
So we're left with 4 plus 2 pi.

00:12:17.440 --> 00:12:23.290
So just like that we were able
to figure out the area of this

00:12:23.290 --> 00:12:25.520
first curvy wall here,
and frankly, that's

00:12:25.520 --> 00:12:26.710
the hardest part.

00:12:26.710 --> 00:12:29.760
Now let's figure out the
area of this curve.

00:12:29.760 --> 00:12:31.220
And actually you're going to
find out that these other

00:12:31.220 --> 00:12:33.830
curves as they go along the
axes are much, much, much

00:12:33.830 --> 00:12:35.760
easier, but we're going to
have to find different

00:12:35.760 --> 00:12:37.480
parametrizations for this.

00:12:37.480 --> 00:12:43.390
So if we take this curve
right here, let's do a

00:12:43.390 --> 00:12:44.590
parametrization for that.

00:12:44.590 --> 00:12:45.490
Actually, you know what?

00:12:45.490 --> 00:12:48.630
Let me continue this in the
next video because I realize

00:12:48.630 --> 00:12:49.920
I've been running
a little longer.

00:12:49.920 --> 00:12:53.110
I'll do the next two walls and
then we'll sum them all up.


Line integral example 1

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.340
The last video was very
abstract in general, and

00:00:03.340 --> 00:00:06.980
I used you know, f of x
and g of t, and h of t.

00:00:06.980 --> 00:00:09.720
What I want to do in this video
is do an actual example.

00:00:09.720 --> 00:00:12.730
Let's say I have f of xy.

00:00:12.730 --> 00:00:17.325
Let's say that f of
xy is equal to xy.

00:00:17.325 --> 00:00:20.595
And let's say that we have
a path in the xy plane, or

00:00:20.595 --> 00:00:21.775
a curve in the xy plane.

00:00:21.775 --> 00:00:29.480
And I'm going to define my
curve by x being equal to

00:00:29.480 --> 00:00:36.430
cosine of t, and y being
equal to sine of t.

00:00:36.430 --> 00:00:40.090
And we're going to go from--
you know, we have to define

00:00:40.090 --> 00:00:43.740
what are our boundaries in our
t --and we're going to go from

00:00:43.740 --> 00:00:48.800
t is equal to 0-- or t is going
to be greater than or equal to

00:00:48.800 --> 00:00:50.930
0 --and then less
than or equal to.

00:00:50.930 --> 00:00:53.340
We're going to deal in
radians, pi over 2.

00:00:53.340 --> 00:00:56.370
If this was degrees, that
would be 90 degrees.

00:00:56.370 --> 00:00:57.440
So that's our curve.

00:00:57.440 --> 00:00:59.860
And immediately you might
already know what this type

00:00:59.860 --> 00:01:00.770
of a curve looks like.

00:01:00.770 --> 00:01:02.900
And I'm going to draw that
really fast right here in and

00:01:02.900 --> 00:01:04.110
we'll try to visualize this.

00:01:04.110 --> 00:01:06.120
I've actually drafted
ahead of time so that

00:01:06.120 --> 00:01:07.590
we can visualize this.

00:01:07.590 --> 00:01:09.530
So this curve right here, if I
were to just draw it in the

00:01:09.530 --> 00:01:13.530
standard xy plane-- do that in
a different color so we can

00:01:13.530 --> 00:01:18.430
make the curve green; let's say
that is y, and this is right

00:01:18.430 --> 00:01:24.400
here x --so when t is equal
to 0, x is going to be

00:01:24.400 --> 00:01:26.310
equal to cosine of 0.

00:01:26.310 --> 00:01:30.620
Cosine of 0 is 1, y is going to
be equal to sine of 0, which

00:01:30.620 --> 00:01:33.500
is 0, so t is equal to 0.

00:01:33.500 --> 00:01:37.650
We're going to be at x equal to
1, that's cosine of 0, and y is

00:01:37.650 --> 00:01:39.706
sine of 0, or y is going to
be 0, so we're going

00:01:39.706 --> 00:01:40.500
to be right there.

00:01:40.500 --> 00:01:44.380
That's what t is equal
to; t is equal to 0.

00:01:44.380 --> 00:01:46.990
When t is equal to pi of 2,
what's going to happen?

00:01:46.990 --> 00:01:50.000
Cosine of pie over 2--
that's the angle; cosine

00:01:50.000 --> 00:01:53.270
of pi over 2 --is 0.

00:01:53.270 --> 00:01:55.010
Sine of pi over 2 is 1.

00:01:55.010 --> 00:01:57.270
We're going to be
at the point 0, 1.

00:01:57.270 --> 00:02:01.410
So this is when we're at
t is equal to pi over 2.

00:02:01.410 --> 00:02:03.340
You might recognize what we're
going to draw is actually the

00:02:03.340 --> 00:02:07.080
first quadrant of the unit
circle; when t is equal to pi

00:02:07.080 --> 00:02:11.530
over 4, or 45 degrees, we're
going to be at square root

00:02:11.530 --> 00:02:12.240
of 2, square root of 2.

00:02:12.240 --> 00:02:13.860
You can try it out for
yourself, but we're just

00:02:13.860 --> 00:02:15.360
going to have a curve
that looks like this.

00:02:15.360 --> 00:02:18.050
It's going to be the
top right of a circle,

00:02:18.050 --> 00:02:18.915
of the units circle.

00:02:18.915 --> 00:02:20.280
It's going to have radius 1.

00:02:20.280 --> 00:02:22.670
And we're going to go in that
direction, from t is equal to

00:02:22.670 --> 00:02:25.830
0, to t is equal to pi over 2.

00:02:25.830 --> 00:02:28.300
That's what this
curve looks like.

00:02:28.300 --> 00:02:32.240
But our goal isn't here just to
graph a parametric equation.

00:02:32.240 --> 00:02:36.680
What we want to do is raise a
fence out of this kind of base

00:02:36.680 --> 00:02:38.685
and rise it to this surface.

00:02:38.685 --> 00:02:41.570
So let's see if we can do that
or at least visualize it first,

00:02:41.570 --> 00:02:44.740
and then we'll use the tools
we used in the last video.

00:02:44.740 --> 00:02:51.210
So right here I've graphed this
function, and I've rotated it a

00:02:51.210 --> 00:02:52.230
little bit so you can
see [UNINTELLIGIBLE]

00:02:52.230 --> 00:02:53.010
case.

00:02:53.010 --> 00:02:56.080
This right here-- let me get
some dark colors out --that

00:02:56.080 --> 00:03:01.680
right there is the x axis, that
in the back is the y axis, and

00:03:01.680 --> 00:03:03.560
the vertical axis
is the z axis.

00:03:03.560 --> 00:03:07.330
And this is actually 2, this
is 1 right here, y equal 1

00:03:07.330 --> 00:03:09.310
is right there, so this
is graphed that way.

00:03:09.310 --> 00:03:12.030
So if I were to graphs this
contour in the xy plane, it

00:03:12.030 --> 00:03:14.900
would be under this graph and
it would go like something like

00:03:14.900 --> 00:03:18.110
this--- let me see if I can
draw it --it would look

00:03:18.110 --> 00:03:18.730
something like this.

00:03:18.730 --> 00:03:20.800
This would be on the xy plane.

00:03:20.800 --> 00:03:25.300
This is the same exact graph,
f of x is equal to xy.

00:03:25.300 --> 00:03:30.080
This is f of x; f of
xy is equal to xy.

00:03:30.080 --> 00:03:32.480
That's both of these,
I just rotated it.

00:03:32.480 --> 00:03:37.170
In this situation that right
there is now the x axis.

00:03:37.170 --> 00:03:40.380
I rotated to the left,
you can kind of imagine.

00:03:40.380 --> 00:03:43.080
That right there is the x axis,
that right there is the y

00:03:43.080 --> 00:03:47.710
axis-- it was rotated closer
to me --that's the z axis.

00:03:47.710 --> 00:03:52.630
And then this curve, if I were
to draw in this rotation, is

00:03:52.630 --> 00:03:55.260
going to look like this: when t
is equal to 0, we're at x is

00:03:55.260 --> 00:03:58.420
equal to 1, y is equal to 0,
and it's going to form a unit

00:03:58.420 --> 00:04:03.000
circle, or half or quarter
of a unit circle like that.

00:04:03.000 --> 00:04:05.745
And when t is equal to pi over
2, we're going to get there.

00:04:05.745 --> 00:04:08.590
And what we want to do
is find the area of the

00:04:08.590 --> 00:04:09.680
curtain that's defined.

00:04:09.680 --> 00:04:12.030
So let's see, let's raise
a curtain from this

00:04:12.030 --> 00:04:14.730
curve up to f of xy.

00:04:14.730 --> 00:04:17.930
So if we keep raising walls
from this up to xy, we're

00:04:17.930 --> 00:04:19.929
going to have a wall looks
something like that.

00:04:22.690 --> 00:04:24.590
Let me shade it in, color
it in so it looks a little

00:04:24.590 --> 00:04:26.660
bit more substantive.

00:04:26.660 --> 00:04:28.530
So a wall that looks
something like that.

00:04:28.530 --> 00:04:30.850
If I were to try to do it here
this would be under the

00:04:30.850 --> 00:04:33.560
ceiling, but the wall
look something like

00:04:33.560 --> 00:04:35.680
that right there.

00:04:35.680 --> 00:04:37.440
We want to find
the area of that.

00:04:37.440 --> 00:04:41.400
We want to find the area of
this right here where the base

00:04:41.400 --> 00:04:45.160
is defined by this curves, and
then the ceiling is defined by

00:04:45.160 --> 00:04:48.550
this surface here, xy, which I
graphed and I rotated

00:04:48.550 --> 00:04:50.270
in two situations.

00:04:50.270 --> 00:04:53.940
Now in the last video we came
up with a, well, you could

00:04:53.940 --> 00:04:56.230
argue whether it's simple, but
the is, well, let's just take

00:04:56.230 --> 00:05:00.000
small arc lengths-- change in
arc lengths, and multiply them

00:05:00.000 --> 00:05:05.300
by the height at that point.

00:05:05.300 --> 00:05:07.910
And those small change in arc
lengths, we called them ds,

00:05:07.910 --> 00:05:10.050
and then the height is just
f of xy at that point.

00:05:10.050 --> 00:05:13.600
And we'll take an infinite sum
of these, from t is equal to 0

00:05:13.600 --> 00:05:16.660
to t will equal pi over 2, and
then that should give us

00:05:16.660 --> 00:05:20.640
the area of this wall.

00:05:20.640 --> 00:05:23.510
So we said is, well, to figure
out the area of that we're just

00:05:23.510 --> 00:05:28.670
going to take the integral from
t is equal to o to t is equal

00:05:28.670 --> 00:05:31.750
to pi over 2-- it doesn't make
a lot of sense when I write it

00:05:31.750 --> 00:05:38.935
like this --of f of xy times--
or let me even better, instead

00:05:38.935 --> 00:05:41.300
of writing f of xy, let me just
write the actual function.

00:05:41.300 --> 00:05:43.110
Let's get a little
bit more concrete.

00:05:43.110 --> 00:05:48.710
So f of xy is xy times-- so the
particular xy --times the

00:05:48.710 --> 00:05:50.520
little change in our arc
length at that point.

00:05:50.520 --> 00:05:53.030
I'm going to be very
hand-wavy here.

00:05:53.030 --> 00:05:55.290
This is all a little bit
review of the last video.

00:05:55.290 --> 00:05:58.890
And we figured out in the last
video this change in arc length

00:05:58.890 --> 00:06:04.990
right here, ds, we figured out
that we could rewrite that as

00:06:04.990 --> 00:06:10.250
the square root of the dx
versus-- or the derivative of x

00:06:10.250 --> 00:06:14.480
with respect to t squared
--plus the derivative of y with

00:06:14.480 --> 00:06:19.090
respect to t squared, and
then all of that times dt.

00:06:19.090 --> 00:06:20.790
So I'm just rebuilding
the formula that we

00:06:20.790 --> 00:06:22.590
got in the last video.

00:06:22.590 --> 00:06:27.090
So this expression can be
rewritten as the integral

00:06:27.090 --> 00:06:33.640
from t is equal to 0 to t is
equal to pi over 2 times xy.

00:06:33.640 --> 00:06:35.160
But you know what?

00:06:35.160 --> 00:06:37.170
Right from the get go we
want everything eventually

00:06:37.170 --> 00:06:38.070
be in terms of t.

00:06:38.070 --> 00:06:42.070
So instead of writing x
times y, let's substitute

00:06:42.070 --> 00:06:43.170
the parametric form.

00:06:43.170 --> 00:06:45.720
So instead of x let's
write cosine of t.

00:06:49.180 --> 00:06:50.370
That is x.

00:06:50.370 --> 00:06:53.980
x is equal to cosine
of t on this curve.

00:06:53.980 --> 00:06:56.700
That's how we define x, in
terms of the parameter t.

00:06:56.700 --> 00:06:59.250
And then times y, which
we're saying is sine of t.

00:07:01.830 --> 00:07:05.360
That's our y; all I
did is rewrote xy in

00:07:05.360 --> 00:07:08.310
terms of t times ds.

00:07:08.310 --> 00:07:16.170
ds is this; it's the square
root of the derivative of x

00:07:16.170 --> 00:07:20.690
with respect to t squared plus
the derivative of y with

00:07:20.690 --> 00:07:22.950
respect to t squared.

00:07:22.950 --> 00:07:25.630
All of that times dt.

00:07:25.630 --> 00:07:27.380
And now we just have to find
these two derivatives.

00:07:27.380 --> 00:07:29.670
And it might seem really hard,
but it's very easy for us to

00:07:29.670 --> 00:07:32.180
find the derivative of x with
respect to t and the derivative

00:07:32.180 --> 00:07:33.360
of y with respect to t.

00:07:33.360 --> 00:07:34.860
I can do it right or down here.

00:07:34.860 --> 00:07:38.320
Let me lose our graphs
for a little bit.

00:07:38.320 --> 00:07:42.600
We know that the derivative of
x with respect to t is just

00:07:42.600 --> 00:07:44.620
going to be: what's the
derivative of cosine of t?

00:07:44.620 --> 00:07:48.790
Well, it's minus sine of t.

00:07:48.790 --> 00:07:51.000
And the derivative of
y was respect to t?

00:07:53.510 --> 00:07:55.670
Derivative of a sine of
anything is the cosine

00:07:55.670 --> 00:07:56.125
of that anything.

00:07:56.125 --> 00:07:59.450
So it's cosine of t.

00:07:59.450 --> 00:08:02.330
And we can substitute these
back into this equation.

00:08:02.330 --> 00:08:06.250
So remember, we're just trying
to find the area of this

00:08:06.250 --> 00:08:10.020
curtain that has our curve here
as kind of its base, and

00:08:10.020 --> 00:08:13.700
has this function, this
surface as it's ceiling.

00:08:13.700 --> 00:08:16.460
So we go back down here,
and let me rewrite

00:08:16.460 --> 00:08:17.240
this whole thing.

00:08:17.240 --> 00:08:22.090
So this becomes the integral
from t is equal to o to t is

00:08:22.090 --> 00:08:26.700
equal to pi over 2-- I don't
like this color --of cosine of

00:08:26.700 --> 00:08:32.290
t, sine of t, cosine times
sine-- that's just the xy

00:08:32.290 --> 00:08:34.900
--times ds, which is this
expression right here.

00:08:34.900 --> 00:08:37.110
And now we can write this as--
I'll go switch back to that

00:08:37.110 --> 00:08:41.870
color I don't like --the
derivative of x with respect to

00:08:41.870 --> 00:08:50.170
t is minus sine of t, and we're
going to square it, plus the

00:08:50.170 --> 00:08:54.750
derivative of y with respect to
t, that's cosine of t, and

00:08:54.750 --> 00:08:57.670
we're going to square it-- let
me make my radical a little bit

00:08:57.670 --> 00:09:01.330
bigger --and then all
of that times dt.

00:09:01.330 --> 00:09:03.820
Now this still might seem like
a really hard integral until

00:09:03.820 --> 00:09:07.510
you realize that this right
here, and when you take a

00:09:07.510 --> 00:09:10.420
negative number and you squared
it, this is the same thing.

00:09:10.420 --> 00:09:13.300
Let me rewrite, do this
in the side right here.

00:09:13.300 --> 00:09:23.260
Minus sine of t squared plus
the cosine of t squared, this

00:09:23.260 --> 00:09:31.310
is equivalent to sine of t
squared plus cosine

00:09:31.310 --> 00:09:32.820
of t squared.

00:09:32.820 --> 00:09:35.090
You lose the sign information
when you square something;

00:09:35.090 --> 00:09:36.430
it just becomes a positive.

00:09:36.430 --> 00:09:37.850
So these two things
are equivalent.

00:09:37.850 --> 00:09:39.640
And this is the most
basic trig identity.

00:09:39.640 --> 00:09:42.690
This comes straight out of the
unit circle definition: sine

00:09:42.690 --> 00:09:45.600
squared plus cosine squared,
this is just equal to 1.

00:09:45.600 --> 00:09:48.020
So all this stuff under
the radical sign is

00:09:48.020 --> 00:09:49.710
just equal to 1.

00:09:49.710 --> 00:09:53.130
And we're taking the square
root of 1 which is just 1.

00:09:53.130 --> 00:09:59.120
So all of this stuff right
here will just become 1.

00:09:59.120 --> 00:10:02.890
And so this whole crazy
integral simplifies a good bit

00:10:02.890 --> 00:10:08.240
and just equals the square root
of t equals 0 to t is equal to

00:10:08.240 --> 00:10:12.540
pi over 2 of-- and I'm going to
switch these around just

00:10:12.540 --> 00:10:14.440
because it will make it a
little easier in the next step

00:10:14.440 --> 00:10:23.440
--of sine of t times
cosine of t, dt.

00:10:23.440 --> 00:10:26.060
All I did, this whole thing
equals 1, got rid of it,

00:10:26.060 --> 00:10:27.340
and I just switched
the order of that.

00:10:27.340 --> 00:10:30.050
It'll make the next up a
little bit easier to explain.

00:10:30.050 --> 00:10:32.920
Now this integral-- You say
sine times cosine, what's

00:10:32.920 --> 00:10:34.350
the antiderivative of that?

00:10:34.350 --> 00:10:38.080
And the first thing you should
recognize is, hey, I have a

00:10:38.080 --> 00:10:41.790
function or an expression here,
and I have its derivative.

00:10:41.790 --> 00:10:44.040
The derivative of
sine is cosine of t.

00:10:44.040 --> 00:10:46.460
So you might be able to a u
substitution in your head;

00:10:46.460 --> 00:10:48.020
it's a good skill to be
able to do in your head.

00:10:48.020 --> 00:10:49.620
But I'll do it very
explicitly here.

00:10:49.620 --> 00:10:51.040
So if you have something
that's derivative, you

00:10:51.040 --> 00:10:52.890
define that something as u.

00:10:52.890 --> 00:11:00.160
So you say u is equal to sine
of t and then du, dt, the

00:11:00.160 --> 00:11:03.650
derivative of u with respect
to t is equal to cosine of t.

00:11:03.650 --> 00:11:07.190
Or if you multiply both sides
by the differential dt, if

00:11:07.190 --> 00:11:09.960
we're not going to be too
rigorous, you get du is

00:11:09.960 --> 00:11:14.140
equal to cosine of t, dt.

00:11:14.140 --> 00:11:16.720
And notice right
here I have a u.

00:11:16.720 --> 00:11:20.860
And then cosine of t, dt,
this thing right here, that

00:11:20.860 --> 00:11:22.470
thing is equal to d of u.

00:11:22.470 --> 00:11:25.030
And then we just have to
redefine the boundaries.

00:11:25.030 --> 00:11:29.350
When t is equal to 0-- I mean
so this thing is going to turn

00:11:29.350 --> 00:11:33.760
into the integral --instead of
t is equal to 0, when t is

00:11:33.760 --> 00:11:35.550
equal to 0 what is u equal to?

00:11:35.550 --> 00:11:40.280
Sine of 0 is 0, so this
goes from u is equal to 0.

00:11:40.280 --> 00:11:44.850
When t is pie over 2
sine of pi over 2 is 1.

00:11:44.850 --> 00:11:47.260
So when t is pi over
2, u is equal to 1.

00:11:47.260 --> 00:11:49.940
So from is equal to 0
to u is equal to 1.

00:11:49.940 --> 00:11:53.000
Just redid the boundaries
in terms of u.

00:11:53.000 --> 00:11:55.280
And then we have instead
of sign of t, I'm

00:11:55.280 --> 00:11:56.930
going to write u.

00:11:56.930 --> 00:12:00.810
And instead of cosine of t, dt,
I'm just going to write du.

00:12:00.810 --> 00:12:04.060
And then this is a super-easy
integral in terms of u.

00:12:04.060 --> 00:12:09.320
This is just equal to: the
antiderivative of u is u 1/2

00:12:09.320 --> 00:12:12.820
times u squared-- we just
raised the exponent and then

00:12:12.820 --> 00:12:16.970
divided by that raised exponent
--so 1/2 u squared, and we're

00:12:16.970 --> 00:12:19.640
going to evaluate
it from 0 to 1.

00:12:19.640 --> 00:12:25.570
And so this is going to be
equal to 1/2 times 1 squared

00:12:25.570 --> 00:12:32.140
minus 1/2 times 0 squared,
which is equal to 1/2 times 1

00:12:32.140 --> 00:12:34.820
minus 0, which is equal to 1/2.

00:12:34.820 --> 00:12:37.750
So we did all that work and
we got a nice simple answer.

00:12:37.750 --> 00:12:42.630
The area of this a curtain-- we
just performed a line integral

00:12:42.630 --> 00:12:48.430
--the area of this curtain
along this curve right here

00:12:48.430 --> 00:12:53.380
is-- let me do it in a
darker color --on 1/2.

00:12:53.380 --> 00:12:55.080
You know, if this was in
centimeters, it would be

00:12:55.080 --> 00:12:56.540
1/2 centimeters squared.

00:12:56.540 --> 00:12:59.710
So I think that was you know,
a pretty neat application

00:12:59.710 --> 00:13:01.460
of the line integral.


Introduction to the line integral

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.760
If we're just dealing with two
dimensions, and we want to find

00:00:03.760 --> 00:00:07.460
the area under a curve, we have
good tools in our toolkit

00:00:07.460 --> 00:00:11.050
already to do it, and I'll
just remind us of our tools.

00:00:11.050 --> 00:00:14.430
so let's say, that's the
x-axis, that's the y-axis, let

00:00:14.430 --> 00:00:18.450
me draw some arbitrary function
right here, and that's

00:00:18.450 --> 00:00:20.130
my function f of x.

00:00:20.130 --> 00:00:25.210
And let's say we want to find
the area between x is equal to

00:00:25.210 --> 00:00:29.810
a, so that's x equal to
a, and x is equal to b.

00:00:29.810 --> 00:00:32.140
We saw this many, many,
many videos ago.

00:00:32.140 --> 00:00:35.010
The way you can think about it,
is you take super small widths

00:00:35.010 --> 00:00:37.610
of x, or super small
changes in x.

00:00:37.610 --> 00:00:40.350
We could call them delta x's,
but because they're so small,

00:00:40.350 --> 00:00:43.650
we're going to call them a dx.

00:00:43.650 --> 00:00:46.300
Super, infinitesimally
small changes in x.

00:00:46.300 --> 00:00:48.780
And then you multiply them
times the value of f

00:00:48.780 --> 00:00:50.300
of x at that point.

00:00:50.300 --> 00:00:52.710
So you multiply it times the
height at that point, which

00:00:52.710 --> 00:00:54.730
is the value of f of x.

00:00:54.730 --> 00:01:01.210
So you get f of x times each of
these infinitesimally small

00:01:01.210 --> 00:01:04.930
bases, that'll give you the
area of this infinitesimally

00:01:04.930 --> 00:01:07.500
narrow rectangle right there.

00:01:07.500 --> 00:01:10.220
And since each of these guys
are infinitely small, you're

00:01:10.220 --> 00:01:12.370
going to have an infinite
number of these rectangles

00:01:12.370 --> 00:01:14.050
in order to fill the space.

00:01:14.050 --> 00:01:15.480
You're going to have
an infinite number

00:01:15.480 --> 00:01:16.880
of these, right?

00:01:16.880 --> 00:01:20.250
And so the tool we use was
the definite integral.

00:01:20.250 --> 00:01:23.970
The definite integral is a sum,
is an infinite sum of these

00:01:23.970 --> 00:01:27.000
infinitely small areas,
or these infinitely

00:01:27.000 --> 00:01:27.910
small rectangles.

00:01:27.910 --> 00:01:31.790
And the notations that we
use, they would go from a b.

00:01:31.790 --> 00:01:34.690
And we've done many videos
on how do you evaluate

00:01:34.690 --> 00:01:35.230
these things.

00:01:35.230 --> 00:01:37.010
I just want to remind
you, conceptually,

00:01:37.010 --> 00:01:38.210
what this is saying.

00:01:38.210 --> 00:01:42.000
This is conceptually saying,
let's take a small change in x,

00:01:42.000 --> 00:01:47.620
multiply it times the height at
that point, and you're going to

00:01:47.620 --> 00:01:49.740
have an infinite number of
these, because these x's are

00:01:49.740 --> 00:01:52.110
super small, they're infinitely
small, so you're going to have

00:01:52.110 --> 00:01:53.330
an infinite number of those.

00:01:53.330 --> 00:01:55.880
So take an infinite sum of all
of those, from x is equal

00:01:55.880 --> 00:01:57.630
to a to x is equal to b.

00:01:57.630 --> 00:02:00.150
And that's just our standard
definite integral.

00:02:00.150 --> 00:02:03.230
Now what I want to do in this
video is extend this, broaden

00:02:03.230 --> 00:02:06.320
this a little bit, to solve, I
guess it maybe could say a

00:02:06.320 --> 00:02:09.240
harder or a broader
class of problems.

00:02:09.240 --> 00:02:13.380
Let's say that we are, let's
go to three dimensions now.

00:02:13.380 --> 00:02:14.970
And I'll just draw
the x-y plane first.

00:02:14.970 --> 00:02:19.820
Maybe I'll keep this, just to
kind of make the analogy clear.

00:02:19.820 --> 00:02:21.450
I'm going to kind of
flatten this, so we

00:02:21.450 --> 00:02:22.460
have some perspective.

00:02:22.460 --> 00:02:25.490
So let's say that this right
here is the y-axis, kind of

00:02:25.490 --> 00:02:27.750
going behind the screen.

00:02:27.750 --> 00:02:29.410
You can imagine if I
just pushed on this

00:02:29.410 --> 00:02:30.600
and knocked it down.

00:02:30.600 --> 00:02:37.480
So that's the y-axis, and that
is my x-axis right there.

00:02:37.480 --> 00:02:41.630
And let's say I some
path in the x-y plane.

00:02:41.630 --> 00:02:44.460
And in order to really define a
path in the x-y plane, I'll

00:02:44.460 --> 00:02:50.560
have to parameterize both
the x and y variables.

00:02:50.560 --> 00:02:55.120
So let's say that x is equal
to, let me switch colors.

00:02:55.120 --> 00:02:57.350
I'm using that orange too much.

00:02:57.350 --> 00:03:03.050
Let's say that x is is equal to
some function of some parameter

00:03:03.050 --> 00:03:08.450
t, and let's say y is equal to
some other function of that

00:03:08.450 --> 00:03:11.320
same parameter t, and let's say
we're going to start, we're

00:03:11.320 --> 00:03:15.090
going to have t go from, t is
going to be greater than or

00:03:15.090 --> 00:03:18.840
equal to a, and then less
than or equal to b.

00:03:18.840 --> 00:03:22.880
Now this will define a path in
the x-y plane, and if this

00:03:22.880 --> 00:03:25.890
seems confusing, you might want
to review the videos on

00:03:25.890 --> 00:03:27.430
parametric equations.

00:03:27.430 --> 00:03:32.060
But essentially, when t is
equal to a, you're going to

00:03:32.060 --> 00:03:35.320
have x is equal to, so t is
equal to a, you're going to

00:03:35.320 --> 00:03:39.630
have x is equal to g of a,
and you're going to have

00:03:39.630 --> 00:03:43.166
y is equal to h of a.

00:03:43.166 --> 00:03:45.580
So you're going to have this
point right here, so maybe it

00:03:45.580 --> 00:03:49.760
might be, I don't know, I'll
just draw a random point here.

00:03:49.760 --> 00:03:52.970
When t is equal to a,
you're going to plot the

00:03:52.970 --> 00:03:54.590
coordinate point g of a.

00:03:54.590 --> 00:03:56.270
That's going to be
our x-coordinate.

00:03:56.270 --> 00:03:58.030
This is g of a, right here.

00:03:58.030 --> 00:04:01.320
And then our y-coordinate
is going to be h of a.

00:04:01.320 --> 00:04:01.520
Right?

00:04:01.520 --> 00:04:03.930
You just put t is equal to a
in each of these equations,

00:04:03.930 --> 00:04:05.430
and then you get a
value for x and y.

00:04:05.430 --> 00:04:08.450
So this coordinate right
here would be h of a.

00:04:08.450 --> 00:04:10.940
And then, you would keep
incrementing t larger and

00:04:10.940 --> 00:04:13.190
larger, until you get to b, but
you're going to get a series of

00:04:13.190 --> 00:04:17.190
points that are going to
look something like that.

00:04:17.190 --> 00:04:22.410
That right there is a curve, or
it's a path, in the x-y plane.

00:04:22.410 --> 00:04:24.430
And you know, you're saying,
how does that relate

00:04:24.430 --> 00:04:25.430
to that right now?

00:04:25.430 --> 00:04:26.250
What are we doing?

00:04:26.250 --> 00:04:28.000
Well, let me just write a c
here, for saying, that's our

00:04:28.000 --> 00:04:29.750
curve, our that's our path.

00:04:29.750 --> 00:04:34.090
Now, let's say I have another
function that associates every

00:04:34.090 --> 00:04:37.010
point in the x-y plane
with some value.

00:04:37.010 --> 00:04:42.480
So let's say I have some
function, f of x y.

00:04:42.480 --> 00:04:44.800
What it does is associate
every point on the x-y

00:04:44.800 --> 00:04:46.170
plane with some value.

00:04:46.170 --> 00:04:47.890
So let me plot f of x y.

00:04:47.890 --> 00:04:50.360
Let me make a
vertical axis here.

00:04:50.360 --> 00:04:53.460
We could do a different color.

00:04:53.460 --> 00:04:56.090
Call it the f of x y axis,
maybe we could even call it

00:04:56.090 --> 00:04:58.010
the z-axis, if you want to.

00:04:58.010 --> 00:05:00.080
But some vertical
axis right there.

00:05:00.080 --> 00:05:04.050
And for every point, so if you
give me an x and a y, and put

00:05:04.050 --> 00:05:07.860
into my f of x y function, it's
going to give you some point.

00:05:07.860 --> 00:05:11.160
So I can just draw some
type of a surface that

00:05:11.160 --> 00:05:12.375
f of x y represents.

00:05:12.375 --> 00:05:14.920
And this'll all become a lot
more concrete when I do

00:05:14.920 --> 00:05:16.440
some concrete examples.

00:05:16.440 --> 00:05:18.950
So let's say that f of x y
looks something like this.

00:05:18.950 --> 00:05:20.890
I'm going to try my
best to draw it.

00:05:20.890 --> 00:05:23.440
I'll do a different color.

00:05:23.440 --> 00:05:25.090
Let's say f of x y.

00:05:25.090 --> 00:05:25.730
Some surface.

00:05:25.730 --> 00:05:26.795
I'll draw part of it.

00:05:26.795 --> 00:05:29.230
It's some surface that
looks, let's say it looks

00:05:29.230 --> 00:05:31.220
something like that.

00:05:31.220 --> 00:05:34.680
That is f of x y.

00:05:34.680 --> 00:05:38.310
And remember, all this is, is
you give me an x, you give me a

00:05:38.310 --> 00:05:41.800
y, you pop it into f of x y,
it's going to give me some

00:05:41.800 --> 00:05:44.060
third value that we're going to
plot in this vertical

00:05:44.060 --> 00:05:44.780
axis right here.

00:05:44.780 --> 00:05:46.760
I mean, example, f of x y?

00:05:46.760 --> 00:05:49.110
It could be, I'm not saying
this is a particular case,

00:05:49.110 --> 00:05:50.650
it could be x plus y.

00:05:50.650 --> 00:05:52.300
It could be f of x y.

00:05:52.300 --> 00:05:53.200
These are just examples.

00:05:53.200 --> 00:05:55.750
It could be x times y.

00:05:55.750 --> 00:05:59.350
If x is 1, y is 2, f of
x y will be 1 times 2.

00:05:59.350 --> 00:06:03.110
But let's say when you plot,
for every point on the x-y

00:06:03.110 --> 00:06:05.570
plane, when you plot f of x y
you get this surface up

00:06:05.570 --> 00:06:08.600
here, and we want to do
something interesting.

00:06:08.600 --> 00:06:12.130
We want to figure out, not the
area under this curve, this

00:06:12.130 --> 00:06:14.230
was very simple when we
did it the first time.

00:06:14.230 --> 00:06:19.640
I want to find the area if you
imagine a curtain, or a fence,

00:06:19.640 --> 00:06:22.570
that goes along this curve.

00:06:22.570 --> 00:06:24.750
You can imagine this being a
very straight linear path,

00:06:24.750 --> 00:06:27.640
going just along the
x-axis from a to b.

00:06:27.640 --> 00:06:30.330
Now we have this kind of crazy,
curvy path that's going

00:06:30.330 --> 00:06:32.070
along the x-y plane.

00:06:32.070 --> 00:06:35.030
And you can imagine if you drew
a wall, or curtain, or a fence

00:06:35.030 --> 00:06:39.750
that went straight up from this
to my f of x y, let me do my

00:06:39.750 --> 00:06:42.100
best effort to draw that.

00:06:42.100 --> 00:06:42.790
Let me draw it.

00:06:42.790 --> 00:06:45.550
So it's going to go up to
there, and maybe this point

00:06:45.550 --> 00:06:47.140
corresponds to there.

00:06:47.140 --> 00:06:49.480
And when you draw that curtain
up, it's going to intersect

00:06:49.480 --> 00:06:51.720
it something like that.

00:06:51.720 --> 00:06:52.940
Let's say it looks
something like that.

00:06:52.940 --> 00:06:56.550
So this point right here
corresponds to that

00:06:56.550 --> 00:06:58.080
point right there.

00:06:58.080 --> 00:07:01.630
So if you imagine, you have a
curtain, f of x y is the roof,

00:07:01.630 --> 00:07:04.620
and this is a, what I've drawn
here, this curve, this kind of

00:07:04.620 --> 00:07:07.600
shows you the bottom of a wall.

00:07:07.600 --> 00:07:09.660
This is some kind
of crazy wall.

00:07:09.660 --> 00:07:13.250
And let me say, this point it
corresponds to, well, actually,

00:07:13.250 --> 00:07:15.205
let me draw it little
bit different.

00:07:18.620 --> 00:07:22.270
This point will correspond to
some point up here, so when you

00:07:22.270 --> 00:07:24.800
trace where it intersects, it
will look something maybe

00:07:24.800 --> 00:07:26.520
like that, I don't know.

00:07:26.520 --> 00:07:27.410
Something like that.

00:07:27.410 --> 00:07:31.220
And I'm trying my best to
help you visualize this.

00:07:31.220 --> 00:07:34.836
So maybe I'll shade this in
to make it a little solid,

00:07:34.836 --> 00:07:36.830
let's say f of x y is
little transparent.

00:07:36.830 --> 00:07:37.500
You can see.

00:07:37.500 --> 00:07:40.360
But you have this
curvy-looking wall here.

00:07:40.360 --> 00:07:43.880
And the whole point of this
video is, how can we figure out

00:07:43.880 --> 00:07:49.890
the area of this curvy-looking
wall, that's essentially the

00:07:49.890 --> 00:07:52.475
wall or the fence that happens
if you go from this curve

00:07:52.475 --> 00:07:56.166
and jump up, and hit the
ceiling at this f of x y?

00:07:56.166 --> 00:07:58.770
So let's think a little bit
about how we can do it.

00:07:58.770 --> 00:08:01.470
Well, if we just use the
analogy of what we did

00:08:01.470 --> 00:08:02.860
previously, we could
say, well look.

00:08:06.740 --> 00:08:11.120
Let's make a little change
in distance of our curve.

00:08:11.120 --> 00:08:13.890
Let's call that ds.

00:08:13.890 --> 00:08:15.740
That's a little change
in distance of my

00:08:15.740 --> 00:08:17.060
curve, right there.

00:08:17.060 --> 00:08:20.200
And if I multiply that change
in distance of the curve times

00:08:20.200 --> 00:08:29.270
f of x y at that point, I'm
going to get the area of that

00:08:29.270 --> 00:08:30.860
little rectangle right there.

00:08:30.860 --> 00:08:31.190
Right?

00:08:31.190 --> 00:08:34.550
So if I take the ds, my change
in my, you can imagine the arc

00:08:34.550 --> 00:08:38.390
length of this curve at that
point, so let me write, you

00:08:38.390 --> 00:08:51.330
know, ds is equal to super
small change in arc length of

00:08:51.330 --> 00:08:52.880
our path, or of our curve.

00:08:52.880 --> 00:08:54.200
That's our ds.

00:08:54.200 --> 00:08:57.340
So you can imagine, the area of
that little rectangle right

00:08:57.340 --> 00:09:02.640
there, along my curvy wall, is
going to be ds, I'll make it a

00:09:02.640 --> 00:09:07.000
capital S, ds times the
height at that point.

00:09:07.000 --> 00:09:10.760
Well, that's f of x y.

00:09:10.760 --> 00:09:13.290
And then if I take the sum,
because these are infinitely

00:09:13.290 --> 00:09:16.980
narrow, these ds's have
infinitely small width, if I

00:09:16.980 --> 00:09:21.860
were take the infinite sum of
all of those guys, from t is

00:09:21.860 --> 00:09:28.130
equal to a to t is equal to b,
right, from t is equal to a, I

00:09:28.130 --> 00:09:30.610
keep taking the sum of those
rectangles, to t is equal

00:09:30.610 --> 00:09:34.110
to b, right there, that
will give me my area.

00:09:34.110 --> 00:09:36.370
I'm just using the exact same
logic as I did up there.

00:09:36.370 --> 00:09:38.660
I'm not being very
mathematically rigorous, but I

00:09:38.660 --> 00:09:40.690
want to give you the intuition
of what we're doing.

00:09:40.690 --> 00:09:44.190
We're really just bending the
base of this thing to get a

00:09:44.190 --> 00:09:47.360
curvy wall instead of a
straight, direct wall

00:09:47.360 --> 00:09:48.270
like we had up here.

00:09:48.270 --> 00:09:51.330
But you're saying, Sal, this is
all abstract, and how can I

00:09:51.330 --> 00:09:53.800
even calculate something like
this, this makes no sense to

00:09:53.800 --> 00:09:57.220
me, I have an s here, I have
an x and a y, I have a t,

00:09:57.220 --> 00:09:58.520
what can I do with this?

00:09:58.520 --> 00:10:00.850
And let's see if we can
make some headway.

00:10:00.850 --> 00:10:03.440
And I promise you, when we do
it with a tangible problem,

00:10:03.440 --> 00:10:05.420
the end product of this video
is going to be a little

00:10:05.420 --> 00:10:07.410
bit hairy to look at.

00:10:07.410 --> 00:10:10.030
But when we do it with an
actual problem, it'll actually,

00:10:10.030 --> 00:10:13.310
I think, be very concrete, and
you'll see it's not too

00:10:13.310 --> 00:10:14.060
hard to deal with.

00:10:14.060 --> 00:10:17.790
But let's see if we can get
all of this in terms of t.

00:10:17.790 --> 00:10:21.390
So first of all, let's
focus just on this ds.

00:10:21.390 --> 00:10:25.070
So let me re-pick
up the x-y axis.

00:10:25.070 --> 00:10:27.910
So if I were to reflip the
x-y, let me switch colors,

00:10:27.910 --> 00:10:30.320
this is just getting
a little monotonous.

00:10:30.320 --> 00:10:33.805
So if I were to reflip the x-y
axis like that, actually, let

00:10:33.805 --> 00:10:35.010
me do that with that same
green, so you know we're

00:10:35.010 --> 00:10:38.000
dealing with the same x-y axis.

00:10:38.000 --> 00:10:42.340
So that's my y-axis,
that is my x-axis.

00:10:42.340 --> 00:10:45.550
And so this path right here, if
I were to just draw it straight

00:10:45.550 --> 00:10:48.045
up like this, it would
look something like this.

00:10:51.060 --> 00:10:51.320
Right?

00:10:51.320 --> 00:10:54.040
That's my path, my arc.

00:10:54.040 --> 00:10:56.930
You know, this is when t is
equal to a, so this is t is

00:10:56.930 --> 00:11:00.090
equal to a, this is
t is equal to b.

00:11:00.090 --> 00:11:01.905
Same thing, I just kind
of picked it back up so

00:11:01.905 --> 00:11:03.110
you can visualize it.

00:11:03.110 --> 00:11:07.040
And we say that we have some
change in arc length, let's

00:11:07.040 --> 00:11:08.880
say, let me switch colors.

00:11:08.880 --> 00:11:11.210
Let's say that this
one right here.

00:11:11.210 --> 00:11:13.470
Let's say that's some small
change in arc length, and

00:11:13.470 --> 00:11:15.640
we're calling that ds.

00:11:15.640 --> 00:11:19.200
Now, is there some way to
relate ds to infinitely

00:11:19.200 --> 00:11:21.710
small changes in x or y?

00:11:21.710 --> 00:11:24.730
Well, if we think about it, if
we really-- and this is all a

00:11:24.730 --> 00:11:26.930
little bit hand-wavy, I'm not
being mathematically rigorous,

00:11:26.930 --> 00:11:29.900
but I think it'll give you the
correct intuition-- if you

00:11:29.900 --> 00:11:33.160
imagine this is, you can figure
out the length of ds if you

00:11:33.160 --> 00:11:37.810
know the length of these
super small changes in x and

00:11:37.810 --> 00:11:40.800
super small changes in y.

00:11:40.800 --> 00:11:43.950
So if this distance right here
is ds, infinitesimally small

00:11:43.950 --> 00:11:46.640
change in x, this distance
right here is dy,

00:11:46.640 --> 00:11:50.090
infinitesimally small
change in y, right?

00:11:50.090 --> 00:11:53.210
Then we could figure out ds
from the Pythagorean Theorem.

00:11:53.210 --> 00:11:57.660
You can say that ds is going to
be, it's the hypotenuse of this

00:11:57.660 --> 00:12:04.310
triang.e It's equal to the
square root of dx squared

00:12:04.310 --> 00:12:09.920
plus dy squared.

00:12:09.920 --> 00:12:13.090
So that seems to make things a
little bit, you know, we can

00:12:13.090 --> 00:12:14.830
get rid of the ds
all of a sudden.

00:12:14.830 --> 00:12:18.740
So let's rewrite this little
expression here, using this

00:12:18.740 --> 00:12:21.850
sense of what ds, is really
the square root of dx

00:12:21.850 --> 00:12:23.160
squared plus dy squared.

00:12:23.160 --> 00:12:25.060
And I'm not being very
rigorous, and actually it's

00:12:25.060 --> 00:12:27.360
very hard to be rigorous with
differentials, but intuitively

00:12:27.360 --> 00:12:29.030
I think it makes
a lot of sense.

00:12:29.030 --> 00:12:32.200
So we can say that this
integral, the area of this

00:12:32.200 --> 00:12:36.080
curvy curtain, is going to be
the integral from t is equal to

00:12:36.080 --> 00:12:44.390
a to t is equal to b of f of x
y, instead of writing ds, we

00:12:44.390 --> 00:12:50.980
can write this, times the
square root of dx squared

00:12:50.980 --> 00:12:52.860
plus dy squared.

00:12:52.860 --> 00:12:55.290
Now we at least got rid of this
big capital S, but we still

00:12:55.290 --> 00:12:58.010
haven't solved the problem of,
how do you solve something, you

00:12:58.010 --> 00:13:00.090
know, an integral, a definite
integral that looks like this?

00:13:00.090 --> 00:13:02.290
We have it in terms of t
here, but we only have it in

00:13:02.290 --> 00:13:04.410
terms of x's and y's here.

00:13:04.410 --> 00:13:06.490
So we need to get
everything in terms of t.

00:13:06.490 --> 00:13:09.710
Well, we know x and y are both
functions of t, so we can

00:13:09.710 --> 00:13:11.530
actually rewrite it like this.

00:13:11.530 --> 00:13:18.220
We can rewrite it as from t is
equal to a, to t is equal to b.

00:13:18.220 --> 00:13:21.800
And f of x y, we can write it,
f is a function of x, which is

00:13:21.800 --> 00:13:26.630
a function of t, and f is also
a function of y, which is

00:13:26.630 --> 00:13:28.510
also a function of t.

00:13:28.510 --> 00:13:31.760
So you give me a t, I'll be
able to give you an x or y, and

00:13:31.760 --> 00:13:34.630
once you give me an x or y,
I can figure out what f is.

00:13:34.630 --> 00:13:38.320
So we have that, and then we
have this part right here.

00:13:38.320 --> 00:13:40.100
I'll do it in orange.

00:13:40.100 --> 00:13:45.380
Square root of dx squared
plus dy squared.

00:13:45.380 --> 00:13:49.000
But we still don't have
things in terms of t.

00:13:49.000 --> 00:13:51.360
We need a dt someplace
here in order be able to

00:13:51.360 --> 00:13:52.460
evaluate this integral.

00:13:52.460 --> 00:13:54.020
And we'll see that in the
next video, when I do

00:13:54.020 --> 00:13:54.890
a concrete problem.

00:13:54.890 --> 00:13:58.560
But I really want to give you a
sense for the end product, the

00:13:58.560 --> 00:14:00.570
formula we're going to get
at the end product of this

00:14:00.570 --> 00:14:02.070
video, where it comes from.

00:14:02.070 --> 00:14:05.710
So one thing we can do, is if
we allow ourselves to

00:14:05.710 --> 00:14:09.180
algebraically manipulate
differentials, what we can do

00:14:09.180 --> 00:14:13.120
is let us multiply
and divide by dt.

00:14:13.120 --> 00:14:16.510
So one way to think about it,
you could rewrite, so let me

00:14:16.510 --> 00:14:19.130
just do this orange
part right here.

00:14:19.130 --> 00:14:20.360
Let's do a little
side right here.

00:14:20.360 --> 00:14:23.860
So if you take this orange
part, and write it in pink, and

00:14:23.860 --> 00:14:29.460
you have dx squared, and then
you have plus dy squared, and

00:14:29.460 --> 00:14:34.040
let's say you just multiply
it times dt over dt, right?

00:14:34.040 --> 00:14:36.880
That's a small change in t,
divided by a small change in t.

00:14:36.880 --> 00:14:39.500
That's 1, so of course you
can multiply it by that.

00:14:39.500 --> 00:14:43.640
If we're to bring in this part
inside of the square root sign,

00:14:43.640 --> 00:14:45.170
right, so let me rewrite this.

00:14:45.170 --> 00:14:51.210
This is the same thing as 1
over dt times the square root

00:14:51.210 --> 00:14:57.130
of dx squared plus dy squared,
and then times that dt.

00:14:57.130 --> 00:14:57.420
Right?

00:14:57.420 --> 00:14:58.670
I just wanted to write it
this way to show you I'm

00:14:58.670 --> 00:15:00.040
just multiplying by 1.

00:15:00.040 --> 00:15:02.740
And here, I'm just taking this
dt, writing it there, and

00:15:02.740 --> 00:15:04.190
leaving this over here.

00:15:04.190 --> 00:15:07.130
And now if I wanted to bring
this into the square root sign,

00:15:07.130 --> 00:15:10.510
this is the same thing, this is
equal to, and I'll do it very

00:15:10.510 --> 00:15:13.790
slowly, just to make sure, I'll
allow you to believe that I'm

00:15:13.790 --> 00:15:15.650
not doing anything shady
with the algebra.

00:15:15.650 --> 00:15:18.760
This is the same thing as the
square root of 1 over dt

00:15:18.760 --> 00:15:21.520
squared, let me make the
radical a little bit bigger,

00:15:21.520 --> 00:15:27.980
times dx squared plus
dy squared, and all of

00:15:27.980 --> 00:15:30.410
that times dt, right?

00:15:30.410 --> 00:15:32.250
I didn't do anything, you could
just take the square root of

00:15:32.250 --> 00:15:34.080
this and you'd get 1 over dt.

00:15:34.080 --> 00:15:38.680
And if I just distribute this,
this is equal to the square

00:15:38.680 --> 00:15:46.770
root, and we have our dt at the
end, of dx squared, or we could

00:15:46.770 --> 00:15:55.700
even write, dx over dt squared,
plus dy over dt squared.

00:15:55.700 --> 00:15:58.590
Right? dx squared over dt
squared is just dx over dt

00:15:58.590 --> 00:16:00.840
squared, same thing
with the y's.

00:16:00.840 --> 00:16:03.100
And now all of a sudden,
this starts to look

00:16:03.100 --> 00:16:03.880
pretty interesting.

00:16:03.880 --> 00:16:06.130
Let's substitute this
expression with this one.

00:16:06.130 --> 00:16:07.700
We said that these
are equivalent.

00:16:07.700 --> 00:16:10.000
And I'll switch colors,
just for the sake of it.

00:16:10.000 --> 00:16:11.470
So we have the integral.

00:16:11.470 --> 00:16:13.250
From t is equal to a.

00:16:13.250 --> 00:16:17.810
Let me get our drawing back, if
I-- from t is equal to a to t

00:16:17.810 --> 00:16:26.050
is equal to b of f of x of t
times, or f of x of t and f of,

00:16:26.050 --> 00:16:29.810
or and y of t, they're both
functions of t, and now instead

00:16:29.810 --> 00:16:34.520
of this expression, we can
write the square root of, well,

00:16:34.520 --> 00:16:39.490
what's dx, what's the change in
x with respect to, whatever

00:16:39.490 --> 00:16:40.150
this parameter is?

00:16:40.150 --> 00:16:41.700
What is dx dt?

00:16:41.700 --> 00:16:55.950
dx dt is the same thing
as g prime of t.

00:16:55.950 --> 00:16:57.660
Right? x is a function of t.

00:16:57.660 --> 00:17:00.130
The function I wrote
is g prime of t.

00:17:00.130 --> 00:17:06.400
And then dy dt is same
thing as h prime of t.

00:17:06.400 --> 00:17:08.740
We could say that, you
know, this function of t.

00:17:08.740 --> 00:17:09.690
So I just wanted to
make that clear.

00:17:09.690 --> 00:17:11.760
We know these two functions,
so we can just take their

00:17:11.760 --> 00:17:12.970
derivatives with respect to t.

00:17:12.970 --> 00:17:15.680
But I'm just going to
leave it in that form.

00:17:15.680 --> 00:17:20.550
So the square root, and we take
the derivative of x with

00:17:20.550 --> 00:17:26.030
respect to t squared, plus the
derivative of y with respect to

00:17:26.030 --> 00:17:29.320
t squared, and all
of that times dt.

00:17:29.320 --> 00:17:32.780
And this might look like some
strange and convoluted formula,

00:17:32.780 --> 00:17:35.130
but this is actually something
that we know how to deal with.

00:17:35.130 --> 00:17:39.340
We've now simplified this
strange, you know, this

00:17:39.340 --> 00:17:42.650
arc-length problem, or this
line integral, right?

00:17:42.650 --> 00:17:43.760
That's essentially
what we're doing.

00:17:43.760 --> 00:17:46.760
We're taking an integral over
a curve, or over a line,

00:17:46.760 --> 00:17:50.530
as opposed to just an
interval on the x-axis.

00:17:50.530 --> 00:17:53.690
We've taken the strange line
integral, that's in terms of

00:17:53.690 --> 00:17:56.730
the arc length of the line, and
x's and y's, and we've put

00:17:56.730 --> 00:17:58.170
everything in terms of t.

00:17:58.170 --> 00:18:00.290
And I'm going to show you that
in the next video, right?

00:18:00.290 --> 00:18:02.200
Everything is going to be
expressed in terms of t,

00:18:02.200 --> 00:18:05.350
so this just turns into a
simple, definite integral.

00:18:05.350 --> 00:18:07.030
So hopefully that didn't
confuse you too much.

00:18:07.030 --> 00:18:09.080
I think you're going to see in
the next video that this, right

00:18:09.080 --> 00:18:11.010
here, is actually a
very straightforward

00:18:11.010 --> 00:18:11.850
thing to implement.

00:18:11.850 --> 00:18:14.250
And just to remind you where
it all came from, I think I

00:18:14.250 --> 00:18:15.470
got the parentheses right.

00:18:15.470 --> 00:18:18.860
This right here was just a
change in our arc length.

00:18:18.860 --> 00:18:20.390
That whole thing right
there was just a

00:18:20.390 --> 00:18:21.330
change in arc length.

00:18:21.330 --> 00:18:26.370
And this is just the height of
our function at that point.

00:18:26.370 --> 00:18:28.820
And we're just summing it,
doing an infinite sum of

00:18:28.820 --> 00:18:30.370
infinitely small lengths.

00:18:30.370 --> 00:18:34.050
So this was a change in our
arc length times the height.

00:18:34.050 --> 00:18:36.640
This is going to have an
infinitely narrow width, and

00:18:36.640 --> 00:18:38.120
they're going to take an
infinite number of these

00:18:38.120 --> 00:18:41.300
rectangles to get the area of
this entire fence, or

00:18:41.300 --> 00:18:42.430
this entire curtain.

00:18:42.430 --> 00:18:45.210
And that's what this definite
integral will give us,

00:18:45.210 --> 00:18:48.270
and we'll actually apply
it in the next video.