WEBVTT
00:00:06.861 --> 00:00:07.360
Hi.
00:00:07.360 --> 00:00:09.020
Welcome back to recitation.
00:00:09.020 --> 00:00:11.420
Today we're going to do a
nice little problem involving
00:00:11.420 --> 00:00:13.530
computing the arc
length of a curve.
00:00:13.530 --> 00:00:17.690
So in particular, consider the
curve given by the equation y
00:00:17.690 --> 00:00:20.250
equals x to the 3/2.
00:00:20.250 --> 00:00:24.011
So I have here a kind
of mediocre sketch
00:00:24.011 --> 00:00:25.260
of what that curve looks like.
00:00:25.260 --> 00:00:28.500
You know, it's curving
upwards not quite
00:00:28.500 --> 00:00:31.960
as fast as a parabola would.
00:00:31.960 --> 00:00:36.190
So I'm interested in the piece
of that curve for x between 0
00:00:36.190 --> 00:00:38.420
and 4, which I've drawn here.
00:00:38.420 --> 00:00:41.030
So why don't you take a
minute, pause the video,
00:00:41.030 --> 00:00:43.030
compute the arc length
of this curve, come back,
00:00:43.030 --> 00:00:44.321
and we can compute it together.
00:00:52.370 --> 00:00:52.870
All right.
00:00:52.870 --> 00:00:53.410
Welcome back.
00:00:53.410 --> 00:00:55.868
Hopefully you had some luck
computing this arc length here.
00:00:55.868 --> 00:00:57.990
So let's set about doing it.
00:00:57.990 --> 00:01:02.520
So I'm sure you remember that
in order to compute arc length,
00:01:02.520 --> 00:01:07.694
first you have to compute the
little piece of arc length ds.
00:01:07.694 --> 00:01:09.860
And we have a couple of
different formulas for that.
00:01:09.860 --> 00:01:12.190
So then after that,
you get an integral,
00:01:12.190 --> 00:01:14.390
and then hopefully it's an
integral you can compute.
00:01:14.390 --> 00:01:16.980
So let's remember what ds is.
00:01:16.980 --> 00:01:20.390
So there are a couple of
different ways to remember it.
00:01:20.390 --> 00:01:22.530
One way, that I
like, is to write
00:01:22.530 --> 00:01:29.540
ds equals the square root of
dx squared plus dy squared.
00:01:29.540 --> 00:01:32.390
So this always reminds me
of the Pythagorean theorem,
00:01:32.390 --> 00:01:34.350
so I find it easy to remember.
00:01:34.350 --> 00:01:36.340
And then from
here, it's not very
00:01:36.340 --> 00:01:37.860
hard to get the
other form, which
00:01:37.860 --> 00:01:40.980
is, you can divide through
by a dx squared inside
00:01:40.980 --> 00:01:43.100
and multiply by dx outside.
00:01:43.100 --> 00:01:47.830
So you can also write this
as the square root of 1
00:01:47.830 --> 00:01:55.964
plus dy/dx squared dx.
00:01:55.964 --> 00:01:58.130
And when you write it in
this form-- it's, you know,
00:01:58.130 --> 00:02:00.240
this is the form
that you can actually
00:02:00.240 --> 00:02:03.720
use to integrate it, to actually
compute the value in question.
00:02:03.720 --> 00:02:07.320
So in our case, we have y
as a function of x, right?
00:02:07.320 --> 00:02:10.100
So we just have
to compete dy/dx.
00:02:10.100 --> 00:02:14.060
So y is x to the 3/2, so
dy/dx is easy to compute,
00:02:14.060 --> 00:02:20.730
y prime, dy/dx is
just 3/2 x to the 1/2,
00:02:20.730 --> 00:02:22.950
or 3/2 square root of x.
00:02:22.950 --> 00:02:26.810
So ds, then-- well, we just
have to plug it in there.
00:02:26.810 --> 00:02:33.470
So that means ds is equal to
the square root of 1 plus-- OK.
00:02:33.470 --> 00:02:34.880
So now you have to square this.
00:02:34.880 --> 00:02:38.046
Well, 3/2 squared is just
9/4, and the square root
00:02:38.046 --> 00:02:39.160
of x squared is x.
00:02:39.160 --> 00:02:46.270
So this is 9/4 x dx.
00:02:46.270 --> 00:02:48.530
So this is the thing that
we want to integrate.
00:02:48.530 --> 00:02:50.840
And now you need
bounds of integration.
00:02:50.840 --> 00:02:52.596
So in our case, this is dx.
00:02:52.596 --> 00:02:53.970
We want to integrate
with respect
00:02:53.970 --> 00:02:55.569
to x, so we need bounds on x.
00:02:55.569 --> 00:02:56.610
And luckily we have them.
00:02:56.610 --> 00:02:59.550
We have 0 less than or equal
to x, less than or equal to 4,
00:02:59.550 --> 00:03:00.910
the bounds that we want.
00:03:00.910 --> 00:03:09.090
So the arc length in
question is the integral
00:03:09.090 --> 00:03:18.940
from 0 to 4 of square
root of 1 plus 9/4 x dx.
00:03:18.940 --> 00:03:22.530
Now, this curve has
the property that this
00:03:22.530 --> 00:03:24.489
is an integral we actually
know how to compute.
00:03:24.489 --> 00:03:24.988
Right?
00:03:24.988 --> 00:03:26.020
There's a-- well, OK.
00:03:26.020 --> 00:03:30.522
So I always lose track of
my constants when I do this,
00:03:30.522 --> 00:03:32.230
so I'm going to do an
extra substitution,
00:03:32.230 --> 00:03:33.550
and then it'll be really easy.
00:03:33.550 --> 00:03:37.710
But you know, this is an
integral-- many of you
00:03:37.710 --> 00:03:39.400
can probably do this
one in your heads,
00:03:39.400 --> 00:03:41.050
basically, at this point.
00:03:41.050 --> 00:03:43.370
This is unusual.
00:03:43.370 --> 00:03:46.030
Even most polynomials
that you write down,
00:03:46.030 --> 00:03:47.780
computing their arc
length is really hard.
00:03:47.780 --> 00:03:50.230
You get nasty things popping up.
00:03:50.230 --> 00:03:52.930
So, you know, I
sort of conspired
00:03:52.930 --> 00:03:56.090
to choose a one that
will have a value that we
00:03:56.090 --> 00:03:57.880
can integrate by hand.
00:03:57.880 --> 00:04:00.920
You don't need to resort to
any sort of numerical method.
00:04:00.920 --> 00:04:03.304
But it happens, in this
case, that that did happen,
00:04:03.304 --> 00:04:03.970
and that's nice.
00:04:03.970 --> 00:04:06.620
So we can we can actually write
down what this arc length is.
00:04:06.620 --> 00:04:10.540
So I'm going to do
the substitution, u
00:04:10.540 --> 00:04:16.750
equals 1 plus 9/4 x.
00:04:16.750 --> 00:04:25.150
So with this substitution, I
get that du is equal to 9/4 dx,
00:04:25.150 --> 00:04:27.700
and since I want to
substitute it the other way,
00:04:27.700 --> 00:04:32.950
I could write that
as dx equals 4/9 du.
00:04:32.950 --> 00:04:35.190
And I also need
to change bounds,
00:04:35.190 --> 00:04:41.400
so when x equals 0, that
goes to u, I put the 0 here,
00:04:41.400 --> 00:04:45.550
u is equal to 1 when
x is equal to 4.
00:04:45.550 --> 00:04:46.520
So I put 4 in here.
00:04:46.520 --> 00:04:50.790
That goes to u equals
10, and so, OK.
00:04:50.790 --> 00:04:56.420
With those substitutions,
I get that the arc length
00:04:56.420 --> 00:05:00.910
that I'm interested in is
the integral from 1 to 10
00:05:00.910 --> 00:05:07.950
of 4/9 times the
square root of u du.
00:05:07.950 --> 00:05:08.450
OK.
00:05:08.450 --> 00:05:11.390
And so now this is,
you know, really easy.
00:05:11.390 --> 00:05:15.830
So this is u to the 1/2,
so I integrate that,
00:05:15.830 --> 00:05:20.340
so I'm going to get u to
the 3/2 divided by 3/2.
00:05:20.340 --> 00:05:35.330
So this is 4/9 times u to the
3/2 divided by 3/2 between u
00:05:35.330 --> 00:05:38.061
equals 1 and u equals 10.
00:05:38.061 --> 00:05:38.560
OK.
00:05:38.560 --> 00:05:45.010
So I can divide here, so this
becomes 8/27 is the constant.
00:05:45.010 --> 00:05:56.350
So this is 8 over 27 times 10
to the 3/2 minus 1 to the 3/2,
00:05:56.350 --> 00:05:58.480
is just 1.
00:05:58.480 --> 00:05:59.020
OK.
00:05:59.020 --> 00:06:01.360
So now if you
wanted to, you know,
00:06:01.360 --> 00:06:04.469
get a decimal approximation
for this number,
00:06:04.469 --> 00:06:06.010
you could put this
into a calculator.
00:06:06.010 --> 00:06:07.600
You can also kind
of eyeball what
00:06:07.600 --> 00:06:10.460
this is, because 10,
the square root of 10
00:06:10.460 --> 00:06:13.460
is just a little bigger than
3, so this is, you know,
00:06:13.460 --> 00:06:16.480
bigger than 27, so
this is bigger than 26.
00:06:16.480 --> 00:06:21.151
So this whole thing is probably
about 8 or a little bit larger.
00:06:21.151 --> 00:06:23.150
Probably going to be a
little bit larger than 8,
00:06:23.150 --> 00:06:24.810
would be my guess.
00:06:24.810 --> 00:06:27.071
So that's, you know,
just rough eyeballing.
00:06:27.071 --> 00:06:29.070
Since you're all sitting
in front of a computer,
00:06:29.070 --> 00:06:35.314
I'm sure you can get a more
precise estimate on your own.
00:06:35.314 --> 00:06:35.980
But there we go.
00:06:35.980 --> 00:06:42.610
So very much just applying the
sort of straightforward tools
00:06:42.610 --> 00:06:45.390
that we've developed for
computing arc lengths.
00:06:45.390 --> 00:06:48.960
You know, using our formulas
for the little element
00:06:48.960 --> 00:06:51.620
of arc length, for the
differential of arc length.
00:06:51.620 --> 00:06:53.275
Computing a derivative,
plugging it in.
00:06:53.275 --> 00:06:55.400
And it happens, in this
case, that we got something
00:06:55.400 --> 00:06:57.960
that we can actually
evaluate the resulting
00:06:57.960 --> 00:07:00.630
integral in a nice closed form.
00:07:00.630 --> 00:07:01.989
So I'll stop there.