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PROFESSOR: Today we're going
to continue our discussion
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00:00:25 --> 00:00:26
of parametric curves.
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I have to tell you
about arc length.
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And let me remind me where
we left off last time.
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This is parametric
curves, continued.
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00:00:45 --> 00:00:50
Last time, we talked about the
parametric representation
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for the circle.
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Or one of the parametric
representations for the circle.
17
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Which was this one here.
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00:00:59 --> 00:01:05
And first we noted that
this does parameterize,
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00:01:05 --> 00:01:07
as we say, the circle.
20
00:01:07 --> 00:01:10
That satisfies the
equation for the circle.
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And it's traced
counterclockwise.
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00:01:17 --> 00:01:20
The picture looks like this.
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Here's the circle.
24
00:01:22 --> 00:01:25
And it starts out here at
t = 0 and it gets up to
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here at time t = pi / 2.
26
00:01:31 --> 00:01:41
So now I have to talk to
you about arc length.
27
00:01:41 --> 00:01:43
In this parametric form.
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00:01:43 --> 00:01:46
And the results should be the
same as arc length around
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00:01:46 --> 00:01:48
this circle ordinarily.
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00:01:48 --> 00:01:54
And we start out with
this basic differential
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00:01:54 --> 00:02:00
relationship. dx ^2
is dx ^2 + dy ^2.
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00:02:00 --> 00:02:04
And then I'm going to take the
square root, divide by dt, so
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00:02:04 --> 00:02:08
the rate of change with respect
to t of s is going to
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be the square root.
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00:02:10 --> 00:02:13
Well, maybe I'll write
it without dividing.
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00:02:13 --> 00:02:15
Just write it as ds.
37
00:02:15 --> 00:02:24
So this would be (dx /
dt)^2 + (dy / dt)^2 dt.
38
00:02:24 --> 00:02:27
So this is what you get
formally from this equation.
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00:02:27 --> 00:02:31
If you take its square roots
and you divide by dt squared in
40
00:02:31 --> 00:02:35
the inside, the square root and
you multiply by dt outside.
41
00:02:35 --> 00:02:36
So that those cancel.
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00:02:36 --> 00:02:39
And this is the formal
connection between the two.
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00:02:39 --> 00:02:44
We'll be saying just a few more
words in a few minutes about
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00:02:44 --> 00:02:48
how to make sense of
that rigorously.
45
00:02:48 --> 00:02:55
Alright so that's the set of
formulas for the infinitesimal,
46
00:02:55 --> 00:02:57
the differential of arc length.
47
00:02:57 --> 00:02:59
And so to figure it out,
I have to differentiate
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00:02:59 --> 00:03:02
x with respect to t.
49
00:03:02 --> 00:03:04
And remember x is up here.
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It's defined by a cos t, so
its derivative is - a sin t.
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And similarly, dy
/ dt = a cos t.
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And so I can plug this in.
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And I get the arc length
element, which is the
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00:03:24 --> 00:03:36
square root of )- a sin
t) ^2 (+ a cos t) ^2 dt.
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Which just becomes the square
root of a ^2 dt, or a dt.
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00:03:44 --> 00:03:46
Now, I was about
to divide by t.
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Let me do that now.
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00:03:48 --> 00:03:52
We can also write the rate
of change of arc length
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with respect to t.
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And that's a, in this case.
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00:03:55 --> 00:04:01
And this gets interpreted
as the speed of the
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particle going around.
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So not only, let me trade these
two guys, not only do we
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have the direction is
counterclockwise, but we also
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have that the speed is, if
you like, it's uniform.
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It's constant speed.
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And the rate is a.
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So that's ds / dt.
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Travelling around.
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And that means that we can
play around with the speed.
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And I just want to point out.
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So the standard thing, what
you'll have to get used to,
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and this is a standard
presentation, you'll
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see this everywhere.
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In your physics classes and
your other math classes, if you
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want to change the speed, so a
new speed going around this
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would be, if I set up
the equations this way.
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Now I'm tracing around
the same circle.
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But the speed is going to turn
out to be, if you figure it
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out, there'll be an
extra factor of k.
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00:05:13 --> 00:05:16
So it'll be a k.
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That's what we'll work
out to be the speed.
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00:05:19 --> 00:05:22
Provided k is positive
and a is positive.
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00:05:22 --> 00:05:30
So we're making
these conventions.
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00:05:30 --> 00:05:37
The constants that we're
using are positive.
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Now, that's the first
and most basic example.
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The one that comes
up constantly.
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Now, let me just make those
comments about notation
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that I wanted to make.
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00:05:47 --> 00:05:52
And we've been treating these
squared differentials here for
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00:05:52 --> 00:05:54
a little while and I just want
to pay attention a little
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00:05:54 --> 00:05:57
bit more carefully to
these manipulations.
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And what's allowed
and what's not.
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And what's justified
and what's not.
95
00:06:01 --> 00:06:06
So the basis for this was this
approximate calculation that we
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00:06:06 --> 00:06:11
had, that delta s ^2 was
delta x ^2 + delta y ^2.
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00:06:11 --> 00:06:16
This is how we justified the
arc length formula before.
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00:06:16 --> 00:06:19
And let me just show you that
the formula that I have up
99
00:06:19 --> 00:06:23
here, this basic formula for
arc length in the parametric
100
00:06:23 --> 00:06:26
form, follows just as
the other one did.
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00:06:26 --> 00:06:31
And now I'm going to do it
slightly more rigorously.
102
00:06:31 --> 00:06:34
I do the division really in
disguise before I take the
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00:06:34 --> 00:06:36
limit of the infinitesimal.
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So all I'm really doing
is I'm doing this.
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Dividing through by this,
and sorry this is still
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approximately equal.
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00:06:43 --> 00:06:45
So I'm not dividing by
something that's 0
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00:06:45 --> 00:06:46
or infinitesimal.
109
00:06:46 --> 00:06:49
I'm dividing by
something non-0.
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And here I have (delta x/ delta
t) ^2 + (delta y / delta t) ^2.
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00:06:56 --> 00:07:02
And then in the limit, I have
ds / dt = to the square
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00:07:02 --> 00:07:04
root of this guy.
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00:07:04 --> 00:07:13
Or, if you like, the
square of it, so.
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00:07:13 --> 00:07:16
So it's legal to divide by
something that's almost
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0 and then take the
limit as we go to 0.
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This is really what
derivatives are all about.
117
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That we get a limit here.
118
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As the denominator goes to 0.
119
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Because the numerator's
going to 0 too.
120
00:07:31 --> 00:07:32
So that's the notation.
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00:07:32 --> 00:07:38
And now I want to warn you,
maybe just a little bit,
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00:07:38 --> 00:07:42
about misuses, if you
like, of the notation.
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00:07:42 --> 00:07:45
We don't do absolutely
everything this way.
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00:07:45 --> 00:07:49
This expression that came up
with the squares, you should
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00:07:49 --> 00:07:55
never write it as this.
126
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This, put it on the board
but very quickly, never.
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00:08:01 --> 00:08:02
OK.
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00:08:02 --> 00:08:07
Don't do that.
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00:08:07 --> 00:08:09
We use these square
differentials, but we don't do
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00:08:09 --> 00:08:12
it with these ratios here.
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But there was another place
which is slightly confusing.
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It looks very similar, where
we did use the square of the
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00:08:19 --> 00:08:20
differential in a denominator.
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00:08:20 --> 00:08:22
And I just want to point out
to you that it's different.
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00:08:22 --> 00:08:23
It's not the same.
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00:08:23 --> 00:08:25
And it is OK.
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00:08:25 --> 00:08:31
And that was this one.
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00:08:31 --> 00:08:33
This thing here.
139
00:08:33 --> 00:08:36
This is a second derivative,
it's something else.
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00:08:36 --> 00:08:39
And it's got a dt squared
in the denominator.
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00:08:39 --> 00:08:41
So it looks rather similar.
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00:08:41 --> 00:08:49
But what this represents is
the quantity d / dt ^2.
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00:08:49 --> 00:08:51
And you can see the
squares came in.
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00:08:51 --> 00:08:53
And squared the
two expressions.
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00:08:53 --> 00:08:58
And then there's also
an x over here.
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00:08:58 --> 00:09:00
So that's legal.
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00:09:00 --> 00:09:02
Those are notations
that we do use.
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00:09:02 --> 00:09:04
And we can even calculate this.
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00:09:04 --> 00:09:05
It has a perfectly
good meaning.
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00:09:05 --> 00:09:07
It's the same as the derivative
with respect to t of the
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00:09:07 --> 00:09:12
derivative of x, which we
already know was - sine.
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00:09:12 --> 00:09:17
Sorry, a sine t, I guess.
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00:09:17 --> 00:09:21
Not this example, but
the previous one.
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00:09:21 --> 00:09:21
Up here.
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00:09:21 --> 00:09:24
So the derivative is this
and so I can differentiate
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00:09:24 --> 00:09:26
a second time.
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00:09:26 --> 00:09:29
And I guess - a cosine t.
158
00:09:29 --> 00:09:31
So that's a perfectly
legal operation.
159
00:09:31 --> 00:09:33
Everything in there
makes sense.
160
00:09:33 --> 00:09:39
Just don't use that.
161
00:09:39 --> 00:09:42
There's another really
unfortunate thing, right which
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00:09:42 --> 00:09:45
is that the 2 creeps in
funny places with signs.
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00:09:45 --> 00:09:46
You have sin^2.
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00:09:48 --> 00:09:50
It would be out here,
it comes up here for
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00:09:50 --> 00:09:51
some strange reason.
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00:09:51 --> 00:09:54
This is just because
typographers are lazy or
167
00:09:54 --> 00:09:57
somebody somewhere in the
history of mathematical
168
00:09:57 --> 00:10:00
typography decided to
let the 2 migrate.
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00:10:00 --> 00:10:04
It would be like putting
the 2 over here.
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00:10:04 --> 00:10:07
There's inconsistency
in mathematics right.
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00:10:07 --> 00:10:11
We're not perfect and people
just develop these notations.
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00:10:11 --> 00:10:14
So we have to live with them.
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00:10:14 --> 00:10:20
The ones that people
accept as conventions.
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00:10:20 --> 00:10:23
The next example that I
want to give you is just
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00:10:23 --> 00:10:24
slightly different.
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00:10:24 --> 00:10:29
It'll be a non-constant
speed parameterization.
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00:10:29 --> 00:10:32
Here x = 2 sine t.
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00:10:32 --> 00:10:37
And y = say, cosine t.
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00:10:37 --> 00:10:40
And let's keep track of
what this one does.
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00:10:40 --> 00:10:44
Now, this is a skill which
I'm going to ask you
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00:10:44 --> 00:10:45
about quite a bit.
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00:10:45 --> 00:10:46
And it's one of several skills.
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00:10:46 --> 00:10:49
You'll have to connect
this with some kind of
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00:10:49 --> 00:10:50
rectangular equation.
185
00:10:50 --> 00:10:51
An equation for x and y.
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00:10:51 --> 00:10:54
And we'll be doing a certain
amount of this today.
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00:10:54 --> 00:10:56
In another context.
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00:10:56 --> 00:11:00
Right here, to see the pattern,
we know that the relationship
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00:11:00 --> 00:11:04
we're going to want to use
is that sin^2 + cos^2 = 1.
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00:11:04 --> 00:11:07
So in fact the right thing
to do here is to take
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00:11:07 --> 00:11:11
1/4 x ^2 + y ^2.
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00:11:11 --> 00:11:17
And that's going to turn out
to be sin ^2 t + cos ^2 t.
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00:11:17 --> 00:11:18
Which is 1.
194
00:11:18 --> 00:11:19
So there's the equation.
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00:11:19 --> 00:11:24
Here's the rectangular equation
for this parametric curve.
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00:11:24 --> 00:11:32
And this describes an ellipse.
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00:11:32 --> 00:11:35
That's not the only information
that we can get here.
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00:11:35 --> 00:11:38
The other information that we
can get is this qualitative
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00:11:38 --> 00:11:41
information of where we start,
where we're going,
200
00:11:41 --> 00:11:42
the direction.
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00:11:42 --> 00:11:46
It starts out, I
claim, at t = 0.
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00:11:46 --> 00:11:53
That's when t = = 0, this is (2
sine 0, cosine 0), right? (2
203
00:11:53 --> 00:12:00
sine 0, cosine 0) =
the point (0, 1).
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00:12:00 --> 00:12:02
So it starts up, up here.
205
00:12:02 --> 00:12:05
At (0, 1).
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00:12:05 --> 00:12:08
And then the next little place,
so this is one thing that
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00:12:08 --> 00:12:12
certainly you want to do. t =
pi / 2 is maybe the next
208
00:12:12 --> 00:12:14
easy point to plot.
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00:12:14 --> 00:12:22
And that's going to be (2
sine pi / 2, cosine pi / 2).
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00:12:22 --> 00:12:27
And that's just (2, 0).
211
00:12:27 --> 00:12:31
And so that's over
here somewhere.
212
00:12:31 --> 00:12:34
This is (2, 0).
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00:12:34 --> 00:12:36
And we know it travels
along the ellipse.
214
00:12:36 --> 00:12:39
And we know the minor axis
is 1, and the major axis
215
00:12:39 --> 00:12:43
is 2, so it's doing this.
216
00:12:43 --> 00:12:45
So this is what
happens at t = 0.
217
00:12:45 --> 00:12:48
This is where we
are at t = pi / 2.
218
00:12:48 --> 00:12:51
And it continues all
the way around, etc.
219
00:12:51 --> 00:12:53
To the rest of the ellipse.
220
00:12:53 --> 00:12:57
This is the direction.
221
00:12:57 --> 00:13:09
So this one happens
to be clockwise.
222
00:13:09 --> 00:13:12
Alright, now let's keep
track of its speed.
223
00:13:12 --> 00:13:25
Let's keep track of the speed,
and also the arc length.
224
00:13:25 --> 00:13:32
So the speed is the square
root of the derivatives here.
225
00:13:32 --> 00:13:42
That would be (2 cosine
t) ^2 + (sine t) ^2.
226
00:13:42 --> 00:13:48
And the arc length is what?
227
00:13:48 --> 00:13:50
Well, if we want to go all the
way around, we need to know
228
00:13:50 --> 00:13:53
that that takes a
total of 2 pi.
229
00:13:53 --> 00:13:55
So 0 to 2 pi.
230
00:13:55 --> 00:13:59
And then we have to integrate
ds, which is this expression.
231
00:13:59 --> 00:14:02
Or ds/ dt, dt.
232
00:14:02 --> 00:14:20
So that's the square root of
4 cosine^2 t + sine ^2 t dt.
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00:14:20 --> 00:14:26
The bad news, if you like,
is that this is not an
234
00:14:26 --> 00:14:38
elementary integral.
235
00:14:38 --> 00:14:42
In other words, no matter how
long you try to figure out how
236
00:14:42 --> 00:14:45
to antidifferentiate this
expression, no matter how
237
00:14:45 --> 00:14:50
many substitutions you
try, you will fail.
238
00:14:50 --> 00:14:52
That's the bad news.
239
00:14:52 --> 00:14:58
The good news is this is not
an elementary integral.
240
00:14:58 --> 00:14:59
It's not an
elementary integral.
241
00:14:59 --> 00:15:03
Which means that this is
the answer to a question.
242
00:15:03 --> 00:15:06
Not something that
you have to work on.
243
00:15:06 --> 00:15:11
So if somebody asks you for
this arc length, you stop here.
244
00:15:11 --> 00:15:14
That's the answer, so it's
actually better than it looks.
245
00:15:14 --> 00:15:19
And we'll try to -- I mean,
I don't expect you to know
246
00:15:19 --> 00:15:21
already what all of the
integrals are that
247
00:15:21 --> 00:15:22
are impossible.
248
00:15:22 --> 00:15:24
And which ones are hard
and which ones are easy.
249
00:15:24 --> 00:15:27
So we'll try to coach
you through when you
250
00:15:27 --> 00:15:28
face these things.
251
00:15:28 --> 00:15:31
It's not so easy to decide.
252
00:15:31 --> 00:15:34
I'll give you a few clues, but.
253
00:15:34 --> 00:15:34
OK.
254
00:15:34 --> 00:15:38
So this is the arc length.
255
00:15:38 --> 00:15:42
Now, I want to move on to
the last thing that we did.
256
00:15:42 --> 00:15:44
Last type of thing that
we did last time.
257
00:15:44 --> 00:15:54
Which is the surface area.
258
00:15:54 --> 00:15:55
And yeah, question.
259
00:15:55 --> 00:16:04
STUDENT: [INAUDIBLE]
260
00:16:04 --> 00:16:05
PROFESSOR: The question,
this is a good question.
261
00:16:05 --> 00:16:09
The question is, when you draw
the ellipse, do you not take
262
00:16:09 --> 00:16:11
into account what t is.
263
00:16:11 --> 00:16:16
The answer is that
this is in disguise.
264
00:16:16 --> 00:16:22
What's going on here is we have
a trouble with plotting in the
265
00:16:22 --> 00:16:24
plane what's really happening.
266
00:16:24 --> 00:16:29
So in other words, it's
kind of in trouble.
267
00:16:29 --> 00:16:33
So the point is that we have
two functions of t, not
268
00:16:33 --> 00:16:35
one. x ( t) and y ( t).
269
00:16:35 --> 00:16:38
So one thing that I can do if
I plot things in the plane.
270
00:16:38 --> 00:16:42
In other words, the main point
to make here is that we're not
271
00:16:42 --> 00:16:46
talking about the situation.
y is a function of x.
272
00:16:46 --> 00:16:47
We're out of that realm now.
273
00:16:47 --> 00:16:50
We're somewhere in a different
part of the universe
274
00:16:50 --> 00:16:51
in our thought.
275
00:16:51 --> 00:16:54
And you should drop
this point of view.
276
00:16:54 --> 00:16:56
So this depiction is not
y as a function of x.
277
00:16:56 --> 00:16:59
Well, that's obvious because
there are two values
278
00:16:59 --> 00:17:01
here, as opposed to one.
279
00:17:01 --> 00:17:02
So we're in trouble with that.
280
00:17:02 --> 00:17:05
And we have that background
parameter, and that's
281
00:17:05 --> 00:17:07
exactly why we're using it.
282
00:17:07 --> 00:17:08
This parameter t.
283
00:17:08 --> 00:17:10
So that we can depict
the entire curve.
284
00:17:10 --> 00:17:14
And deal with it as one thing.
285
00:17:14 --> 00:17:17
So since I can't really draw
it, and since t is nowhere on
286
00:17:17 --> 00:17:20
the map, you should sort of
imagine it as time, and there's
287
00:17:20 --> 00:17:22
some kind of trajectory
which is travelling around.
288
00:17:22 --> 00:17:25
And then I just labelled
a couple of the places.
289
00:17:25 --> 00:17:28
If somebody asked you to draw a
picture of this, well, I'll
290
00:17:28 --> 00:17:31
tell you exactly where you
need the picture in just
291
00:17:31 --> 00:17:33
one second, alright.
292
00:17:33 --> 00:17:36
It's going to come up right
now in surface area.
293
00:17:36 --> 00:17:40
But otherwise, if nobody asks
you to, you don't even have to
294
00:17:40 --> 00:17:44
put down t = 0 and
t = pi / 2 here.
295
00:17:44 --> 00:17:46
Because nobody
demanded it of you.
296
00:17:46 --> 00:17:47
Another question.
297
00:17:47 --> 00:17:52
STUDENT: [INAUDIBLE]
298
00:17:52 --> 00:17:54
PROFESSOR: So, another very
good question which is exactly
299
00:17:54 --> 00:17:55
connected to this picture.
300
00:17:55 --> 00:17:58
So how is it that we're going
to use the picture, and how is
301
00:17:58 --> 00:18:02
it we're going to use
the notion of the t.
302
00:18:02 --> 00:18:07
The question was, why is this
from t = 0 to t = 2 pi?
303
00:18:07 --> 00:18:11
That does use the t information
on this diagram. the point is,
304
00:18:11 --> 00:18:13
we do know that t starts here.
305
00:18:13 --> 00:18:16
This is pi / 2, this
is pi, this is 3 pi /
306
00:18:16 --> 00:18:17
2, and this is 2 pi.
307
00:18:17 --> 00:18:19
When you go all the way
around once, it's going
308
00:18:19 --> 00:18:21
to come back to itself.
309
00:18:21 --> 00:18:23
These are periodic
functions of period 2 pi.
310
00:18:23 --> 00:18:26
And they come back to
themselves exactly at 2 pi.
311
00:18:26 --> 00:18:29
And so that's why we know in
order to get around once, we
312
00:18:29 --> 00:18:32
need to go from 0 to 2 pi.
313
00:18:32 --> 00:18:34
And the same thing is going
to come up with surface
314
00:18:34 --> 00:18:35
area right now.
315
00:18:35 --> 00:18:39
That's going to be the issue,
is what range of t we're going
316
00:18:39 --> 00:18:45
to need when we compute
the surface area.
317
00:18:45 --> 00:18:52
STUDENT: [INAUDIBLE]
318
00:18:52 --> 00:18:54
PROFESSOR: In a question, what
you might be asked is what's
319
00:18:54 --> 00:18:57
the rectangular equation
for a parametric curve?
320
00:18:57 --> 00:19:01
So that would be 1/4
x^2 + y ^2 = 1.
321
00:19:01 --> 00:19:03
And then you might
be asked, plot it.
322
00:19:03 --> 00:19:06
Well, that would be a
picture of the ellipse.
323
00:19:06 --> 00:19:09
OK, those are types of
questions that are
324
00:19:09 --> 00:19:10
legal questions.
325
00:19:10 --> 00:19:27
STUDENT: [INAUDIBLE]
326
00:19:27 --> 00:19:29
PROFESSOR: The question
is, do I need to know
327
00:19:29 --> 00:19:30
any specific formulas?
328
00:19:30 --> 00:19:33
Any formulas that you know
and remember will help you.
329
00:19:33 --> 00:19:35
They may be of limited use.
330
00:19:35 --> 00:19:38
I'm not going to ask you to
memorize anything except,
331
00:19:38 --> 00:19:40
I guarantee you that the
circle is going to come up.
332
00:19:40 --> 00:19:43
Not the ellipse, the circle
will come up everywhere
333
00:19:43 --> 00:19:44
in your life.
334
00:19:44 --> 00:19:47
So at least at MIT,
your life at MIT.
335
00:19:47 --> 00:19:52
We're very round here.
336
00:19:52 --> 00:19:52
Yeah, another question.
337
00:19:52 --> 00:19:57
STUDENT: I'm just a tiny bit
confused back to the basics.
338
00:19:57 --> 00:19:58
This is more a question
from yesterday, I guess.
339
00:19:58 --> 00:20:04
But when you have your original
ds^2= dx^2 + dy ^2, and then
340
00:20:04 --> 00:20:10
you integrate that to get arc
length, how are you, the
341
00:20:10 --> 00:20:14
integral has dx's and dy's.
342
00:20:14 --> 00:20:18
So how are you just integrating
with respect to dx?
343
00:20:18 --> 00:20:22
PROFESSOR: OK, the question is
how are we just integrating
344
00:20:22 --> 00:20:24
with respect to x?
345
00:20:24 --> 00:20:26
So this is a question which
goes back to last time.
346
00:20:26 --> 00:20:30
And what is it with
arc length. so.
347
00:20:30 --> 00:20:35
I'm going to have to answer
that question in connection
348
00:20:35 --> 00:20:36
with what we did today.
349
00:20:36 --> 00:20:38
So this is a subtle question.
350
00:20:38 --> 00:20:43
But I want you to realize that
this is actually an important
351
00:20:43 --> 00:20:44
conceptual step here.
352
00:20:44 --> 00:20:49
So shhh, everybody, listen.
353
00:20:49 --> 00:20:53
If you're representing
one-dimensional objects, which
354
00:20:53 --> 00:20:56
are curves, maybe, in space.
355
00:20:56 --> 00:20:58
Or in two dimensions.
356
00:20:58 --> 00:21:01
When you're keeping track of
arc length, you're going to
357
00:21:01 --> 00:21:03
have to have an integral
which is with respect
358
00:21:03 --> 00:21:05
to some variable.
359
00:21:05 --> 00:21:08
But that variable,
you get to pick.
360
00:21:08 --> 00:21:12
And we're launching now into
this variety of choices of
361
00:21:12 --> 00:21:15
variables with respect to which
you can represent something.
362
00:21:15 --> 00:21:18
Now, there are some
disadvantages on the circle
363
00:21:18 --> 00:21:21
to representing things with
respect to the variable x.
364
00:21:21 --> 00:21:24
Because there are two
points on the circle here.
365
00:21:24 --> 00:21:26
On the other hand, you
actually can succeed
366
00:21:26 --> 00:21:27
with half the circle.
367
00:21:27 --> 00:21:29
So you can figure out the
arc length that way.
368
00:21:29 --> 00:21:32
And then you can set it
up as an integral dx.
369
00:21:32 --> 00:21:34
But you can also set it up
as an integral with respect
370
00:21:34 --> 00:21:37
to any parameter you want.
371
00:21:37 --> 00:21:40
And the uniform parameter is
perhaps the easiest one.
372
00:21:40 --> 00:21:43
This one is perhaps
the easiest one.
373
00:21:43 --> 00:21:48
And so now the thing that's
strange about this perspective,
374
00:21:48 --> 00:21:51
and I'm going to make this
point later in the
375
00:21:51 --> 00:21:52
lecture as well.
376
00:21:52 --> 00:21:56
Is that the letters x and y, as
I say, you should drop this
377
00:21:56 --> 00:22:00
notion that y is
a function of x.
378
00:22:00 --> 00:22:03
This is what we're throwing
away at this point.
379
00:22:03 --> 00:22:06
What we're thinking of is, you
can describe things in terms
380
00:22:06 --> 00:22:08
of any coordinate you want.
381
00:22:08 --> 00:22:11
You just have to say what each
one is in terms of the others.
382
00:22:11 --> 00:22:15
And these x and y over here
are where we are in the
383
00:22:15 --> 00:22:18
Cartesian coordinate system.
384
00:22:18 --> 00:22:21
They're not, and in this
case they're functions
385
00:22:21 --> 00:22:24
of some other variable.
386
00:22:24 --> 00:22:25
Some other variable.
387
00:22:25 --> 00:22:27
So they're each functions.
388
00:22:27 --> 00:22:29
So the letters x and y
just changed on you.
389
00:22:29 --> 00:22:33
They mean something different.
x is no longer the variable.
390
00:22:33 --> 00:22:36
It's the function.
391
00:22:36 --> 00:22:39
Right?
392
00:22:39 --> 00:22:40
You're going to have
to get used to that.
393
00:22:40 --> 00:22:42
That's because we
run out of letters.
394
00:22:42 --> 00:22:44
And we kind of want to use
all of them the way we want.
395
00:22:44 --> 00:22:48
I'll say some more
about that later.
396
00:22:48 --> 00:22:51
So now I want to do this
surface area example.
397
00:22:51 --> 00:22:59
I'm going to just take the
surface area of the ellipsoid.
398
00:22:59 --> 00:23:15
The surface of the ellipsoid
formed by revolving
399
00:23:15 --> 00:23:19
this previous example,
which was Example 2.
400
00:23:19 --> 00:23:28
Around the y axis.
401
00:23:28 --> 00:23:30
So we want to set up that
surface area integral
402
00:23:30 --> 00:23:32
here for you.
403
00:23:32 --> 00:23:38
Now, I remind you that the
area element looks like this.
404
00:23:38 --> 00:23:42
If you're revolving around the
y axis, that means you're
405
00:23:42 --> 00:23:43
going around this way
and you have some curve.
406
00:23:43 --> 00:23:44
In this case it's this
piece of an ellipse.
407
00:23:44 --> 00:23:46
If you sweep it around
you're going to get what's
408
00:23:46 --> 00:23:48
called an ellipsoid.
409
00:23:48 --> 00:23:51
And there's a little
chunk here, that you're
410
00:23:51 --> 00:23:53
wrapping around.
411
00:23:53 --> 00:23:58
And the important thing you
need besides this ds, this arc
412
00:23:58 --> 00:24:04
length piece over here, is
the distance to the axis.
413
00:24:04 --> 00:24:06
So that's this horizontal
distance here.
414
00:24:06 --> 00:24:09
I'll draw it in another color.
415
00:24:09 --> 00:24:15
And that horizontal
distance now has a name.
416
00:24:15 --> 00:24:18
And this is, again, the virtue
of this coordinate system.
417
00:24:18 --> 00:24:20
The t is something else.
418
00:24:20 --> 00:24:21
This has a name.
419
00:24:21 --> 00:24:22
This distance has a name.
420
00:24:22 --> 00:24:27
This distance is called x.
421
00:24:27 --> 00:24:29
And it even has a formula.
422
00:24:29 --> 00:24:36
Its formula is 2 sin t.
423
00:24:36 --> 00:24:38
In terms of t.
424
00:24:38 --> 00:24:44
So the full formula up for the
integral here is, I have to
425
00:24:44 --> 00:24:47
take the circumference when
I spin this thing around.
426
00:24:47 --> 00:24:48
And this little arc
length element.
427
00:24:48 --> 00:24:53
So I have here 2 pi ( 2 sin t).
428
00:24:53 --> 00:24:55
That's the x variable here.
429
00:24:55 --> 00:25:00
And then I have here ds,
which is kind of a mess.
430
00:25:00 --> 00:25:04
So unfortunately I don't
quite have room for it.
431
00:25:04 --> 00:25:05
Plan ahead.
432
00:25:05 --> 00:25:13
Square root of 4 cos^2
t + sin^2 t, is that
433
00:25:13 --> 00:25:15
what it was, dt?
434
00:25:15 --> 00:25:17
Alright, I guess I
squeezed it in there.
435
00:25:17 --> 00:25:20
So that was the arc length,
which I re-copied from
436
00:25:20 --> 00:25:21
this board above.
437
00:25:21 --> 00:25:24
That was the ds piece.
438
00:25:24 --> 00:25:29
It's this whole thing
including the dt.
439
00:25:29 --> 00:25:32
That's the answer
except for one thing.
440
00:25:32 --> 00:25:33
What else do we need?
441
00:25:33 --> 00:25:36
We don't just need the
integrand, this is half of
442
00:25:36 --> 00:25:37
setting up an integral.
443
00:25:37 --> 00:25:40
The other half of setting up
an integral is the limits.
444
00:25:40 --> 00:25:42
We need specific limits here.
445
00:25:42 --> 00:25:46
Otherwise we don't have a
number that we can get out.
446
00:25:46 --> 00:25:50
So we now have to think
about what the limits are.
447
00:25:50 --> 00:25:52
And maybe somebody can see.
448
00:25:52 --> 00:25:54
It has something to do
with this diagram of
449
00:25:54 --> 00:25:55
the ellipse over here.
450
00:25:55 --> 00:25:58
Can somebody guess what it is?
451
00:25:58 --> 00:25:59
0 to pi.
452
00:25:59 --> 00:26:02
Well, that was quick.
453
00:26:02 --> 00:26:02
That's it.
454
00:26:02 --> 00:26:05
Because we go from the top
to the bottom, but we don't
455
00:26:05 --> 00:26:06
want to continue around.
456
00:26:06 --> 00:26:08
We don't want to go from 0 to
2 pi, because that would be
457
00:26:08 --> 00:26:12
duplicating what we're going
to get when we spin around.
458
00:26:12 --> 00:26:13
And we know that we start at 0.
459
00:26:13 --> 00:26:16
It's interesting because it
descends when you change
460
00:26:16 --> 00:26:18
variables to think of it in
terms of the y variable it's
461
00:26:18 --> 00:26:20
going the opposite way.
462
00:26:20 --> 00:26:24
But anyway, just one piece
of this is what we want.
463
00:26:24 --> 00:26:27
So that's this setup.
464
00:26:27 --> 00:26:36
And now I claim that this is
actually a doable integral.
465
00:26:36 --> 00:26:37
However, it's long.
466
00:26:37 --> 00:26:39
I'm going to spare you,
I'll just tell you how
467
00:26:39 --> 00:26:41
you would get started.
468
00:26:41 --> 00:26:45
You would use the
substitution u = cos t.
469
00:26:45 --> 00:26:53
And then the du is going
to be - sin t dt.
470
00:26:53 --> 00:26:56
But then, unfortunately,
there's a lot more.
471
00:26:56 --> 00:26:58
There's another trig
substitution with some
472
00:26:58 --> 00:27:01
other multiple of the
cosine and so forth.
473
00:27:01 --> 00:27:02
So it goes on and on.
474
00:27:02 --> 00:27:06
If you want to check
it yourself, you can.
475
00:27:06 --> 00:27:08
There's an inverse trig
substitution which isn't
476
00:27:08 --> 00:27:11
compatible with this one.
477
00:27:11 --> 00:27:17
But it can be done.
478
00:27:17 --> 00:27:22
Calculated.
479
00:27:22 --> 00:27:26
In elementary terms.
480
00:27:26 --> 00:27:31
Yeah, another question.
481
00:27:31 --> 00:27:31
STUDENT: [INAUDIBLE]
482
00:27:31 --> 00:27:34
PROFESSOR: So, if you get this
on an exam, I'm going to have
483
00:27:34 --> 00:27:35
to coach you through it.
484
00:27:35 --> 00:27:37
Either I'm going to have to
tell you don't evaluate it
485
00:27:37 --> 00:27:40
or, you're going to have
to work really hard.
486
00:27:40 --> 00:27:42
Or here's the first step,
and then the next step
487
00:27:42 --> 00:27:44
is, keep on going.
488
00:27:44 --> 00:27:44
Or something.
489
00:27:44 --> 00:27:47
I'll have to give
you some cues.
490
00:27:47 --> 00:27:49
Because it's quite long.
491
00:27:49 --> 00:27:52
This is way too long for an
exam, this particular one.
492
00:27:52 --> 00:27:53
OK.
493
00:27:53 --> 00:27:55
It's not too long
for a problem set.
494
00:27:55 --> 00:27:57
This is where I would leave
you off if I were giving it
495
00:27:57 --> 00:27:58
to you on a problem set.
496
00:27:58 --> 00:28:00
Just to give you an idea of
the order of magnitude.
497
00:28:00 --> 00:28:02
Whereas one of the ones that I
did yesterday, I wouldn't even
498
00:28:02 --> 00:28:11
give you on a problem
set, it was so long.
499
00:28:11 --> 00:28:17
So now, our next job is to
move on to polar coordinates.
500
00:28:17 --> 00:28:20
Now, polar coordinate involve
the geometry of circles.
501
00:28:20 --> 00:28:23
As I said, we really
love circles here.
502
00:28:23 --> 00:28:24
We're very around.
503
00:28:24 --> 00:28:28
Just as I love 0, the rest of
the Institute loves circles.
504
00:28:28 --> 00:28:47
So we're going to
do that right now.
505
00:28:47 --> 00:28:58
What we're going to talk about
now is polar coordinates.
506
00:28:58 --> 00:29:01
Which are set up in
the following way.
507
00:29:01 --> 00:29:04
It's a way of describing
the points in the plane.
508
00:29:04 --> 00:29:07
Here is a point in a plane,
and here's what we think
509
00:29:07 --> 00:29:10
of as the usual x-y axes.
510
00:29:10 --> 00:29:13
And now this point is going to
be described by a different
511
00:29:13 --> 00:29:16
pair of coordinates,
different pair of numbers.
512
00:29:16 --> 00:29:26
Namely, the distance
to the origin.
513
00:29:26 --> 00:29:30
And the second parameter
here, second number here,
514
00:29:30 --> 00:29:32
is this angle theta.
515
00:29:32 --> 00:29:43
Which is the angle of ray
from origin with the
516
00:29:43 --> 00:29:48
horizontal axis.
517
00:29:48 --> 00:29:50
So that's what it
is in language.
518
00:29:50 --> 00:29:53
And you should put this in
quotation marks, because
519
00:29:53 --> 00:29:57
it's not a perfect match.
520
00:29:57 --> 00:30:01
This is geometrically what you
should always think of, but the
521
00:30:01 --> 00:30:06
technical details involve
dealing directly with formulas.
522
00:30:06 --> 00:30:09
The first formula is
the formula for x.
523
00:30:09 --> 00:30:11
And this is the fundamental,
these two are the
524
00:30:11 --> 00:30:12
fundamental ones.
525
00:30:12 --> 00:30:16
Namely, x = r cos theta.
526
00:30:16 --> 00:30:18
The second formula is the
formula for y, which
527
00:30:18 --> 00:30:21
is r sin theta.
528
00:30:21 --> 00:30:25
So these are the
unambiguous definitions
529
00:30:25 --> 00:30:27
of polar coordinates.
530
00:30:27 --> 00:30:28
This is it.
531
00:30:28 --> 00:30:32
And this is the thing from
which all other almost correct
532
00:30:32 --> 00:30:37
statements almost follow.
533
00:30:37 --> 00:30:39
But this is the one you
should trust always.
534
00:30:39 --> 00:30:44
This is the un
ambiguous statement.
535
00:30:44 --> 00:30:47
So let me give you an example
something that's close to
536
00:30:47 --> 00:30:57
being a good formula and is
certainly useful in its way.
537
00:30:57 --> 00:31:02
Namely, you can think of
r as being the square
538
00:31:02 --> 00:31:05
root of x ^2 + y ^2.
539
00:31:05 --> 00:31:07
That's easy enough
to derive, it's the
540
00:31:07 --> 00:31:08
distance to the origin.
541
00:31:08 --> 00:31:11
That's pretty obvious.
542
00:31:11 --> 00:31:14
And the formula for theta,
which you can also derive,
543
00:31:14 --> 00:31:21
which is that it's the
inverse tangent of y / x.
544
00:31:21 --> 00:31:24
However, let me just warn
you that these formulas
545
00:31:24 --> 00:31:26
are slightly ambiguous.
546
00:31:26 --> 00:31:33
So somewhat ambiguous.
547
00:31:33 --> 00:31:35
In other words, you can't
just apply them blindly.
548
00:31:35 --> 00:31:37
You actually have to look
at a picture in order
549
00:31:37 --> 00:31:38
to get them right.
550
00:31:38 --> 00:31:43
In particular, r could
be plus or minus here.
551
00:31:43 --> 00:31:48
And when you take the inverse
tangent, there's an ambiguity
552
00:31:48 --> 00:31:56
between, it's the same as the
inverse tangent of - y / - x.
553
00:31:56 --> 00:32:00
So these minus signs are a
plague on your existence.
554
00:32:00 --> 00:32:05
And you're not going to get a
completely unambiguous answer
555
00:32:05 --> 00:32:07
out of these formulas without
paying attention
556
00:32:07 --> 00:32:08
to the diagram.
557
00:32:08 --> 00:32:10
On the other hand, the
formula up in the box
558
00:32:10 --> 00:32:14
there always works.
559
00:32:14 --> 00:32:15
So when people mean
polar coordinates,
560
00:32:15 --> 00:32:17
they always mean that.
561
00:32:17 --> 00:32:22
And then they have conventions,
which sometimes match things up
562
00:32:22 --> 00:32:27
with the formulas over
on this next board.
563
00:32:27 --> 00:32:32
Let me give you various
examples here first.
564
00:32:32 --> 00:32:36
But maybe first I should
I should draw the two
565
00:32:36 --> 00:32:38
coordinate systems.
566
00:32:38 --> 00:32:40
So the coordinate system
that we're used to is the
567
00:32:40 --> 00:32:43
rectangular coordinate system.
568
00:32:43 --> 00:32:49
And maybe I'll draw it in
orange and green here.
569
00:32:49 --> 00:32:59
So these are the coordinate
lines y = 0, y = 1, y = 2.
570
00:32:59 --> 00:33:01
That's how the coordinate
system works.
571
00:33:01 --> 00:33:08
And over here we have the rest
of the coordinate system.
572
00:33:08 --> 00:33:10
And this is the way we're
thinking of x and y now.
573
00:33:10 --> 00:33:12
We're no longer thinking of y
as a function of x and x as a
574
00:33:12 --> 00:33:15
function of y, we're thinking
of x as a label of a
575
00:33:15 --> 00:33:16
place in a plane.
576
00:33:16 --> 00:33:20
And y as a label of
a place in a plane.
577
00:33:20 --> 00:33:27
So here we have x = 0,
x = 1, x = 2, etc.
578
00:33:27 --> 00:33:30
Here's x = - 1.
579
00:33:30 --> 00:33:31
So forth.
580
00:33:31 --> 00:33:37
So that's what the rectangular
coordinate system looks like.
581
00:33:37 --> 00:33:41
And now I should draw the other
coordinate system that we have.
582
00:33:41 --> 00:33:47
Which is this guy here.
583
00:33:47 --> 00:33:49
Well, close enough.
584
00:33:49 --> 00:33:54
And these guys here.
585
00:33:54 --> 00:33:57
Kind of this bulls-eye
or target operation.
586
00:33:57 --> 00:34:01
And this one is, say,
theta = pi / 2.
587
00:34:01 --> 00:34:03
This is theta = 0.
588
00:34:03 --> 00:34:07
This is theta = - pi / 4.
589
00:34:07 --> 00:34:11
For instance, so I've just
labeled for you three of
590
00:34:11 --> 00:34:17
the rays on this diagram.
591
00:34:17 --> 00:34:23
It's kind of like
a radar screen.
592
00:34:23 --> 00:34:28
And then in pink, this is
maybe r = 2, the radius 2.
593
00:34:28 --> 00:34:33
And inside is r = 1.
594
00:34:33 --> 00:34:38
So it's a different coordinate
system for the plane.
595
00:34:38 --> 00:34:42
And again, the letter r
represents measuring how far
596
00:34:42 --> 00:34:44
we are from the origin.
597
00:34:44 --> 00:34:47
The theta represents
something about the angle,
598
00:34:47 --> 00:34:50
which ray we're on.
599
00:34:50 --> 00:34:52
And they're just two
different variables.
600
00:34:52 --> 00:35:10
And this is a very different
kind of coordinate system.
601
00:35:10 --> 00:35:15
OK so, our main job is
just to get used to this.
602
00:35:15 --> 00:35:15
For now.
603
00:35:15 --> 00:35:18
You will be using
this a lot in 18.02.
604
00:35:18 --> 00:35:20
It's very useful in physics.
605
00:35:20 --> 00:35:25
And our job is just to
get started with it.
606
00:35:25 --> 00:35:29
And so, let's try a
few examples here.
607
00:35:29 --> 00:35:31
Tons of examples.
608
00:35:31 --> 00:35:34
We'll start out very slow.
609
00:35:34 --> 00:35:41
If you have (x, y) = (1, - 1),
that's a point in the plane.
610
00:35:41 --> 00:35:44
I can draw that point.
611
00:35:44 --> 00:35:46
It's down here, right?
612
00:35:46 --> 00:35:50
This is - 1 and this is 1, and
here's my point, (1, - 1).
613
00:35:50 --> 00:35:54
I can figure out what the
representative is of this
614
00:35:54 --> 00:35:56
in polar coordinates.
615
00:35:56 --> 00:36:03
So in polar coordinates,
there are actually a
616
00:36:03 --> 00:36:05
bunch of choices here.
617
00:36:05 --> 00:36:09
First of all, I'll
tell you one choice.
618
00:36:09 --> 00:36:11
If I start with the angle
horizontally, I wrap
619
00:36:11 --> 00:36:13
all the way around.
620
00:36:13 --> 00:36:19
That would be to this ray here,
let's do it in green again.
621
00:36:19 --> 00:36:22
Alright, I labeled it actually
as - pi / 4, but another way of
622
00:36:22 --> 00:36:27
looking at over here it is
that it's this angle here.
623
00:36:27 --> 00:36:31
So that would be r =
square root of 2.
624
00:36:31 --> 00:36:38
Theta = 7 pi / 4.
625
00:36:38 --> 00:36:41
So that's one possibility of
the angle and the distance.
626
00:36:41 --> 00:36:45
I know the distance is a square
root of 2, that's not hard.
627
00:36:45 --> 00:36:48
Another way of looking at it is
the way which was suggested
628
00:36:48 --> 00:36:51
when I labeled this
with a negative angle.
629
00:36:51 --> 00:36:56
And that would be r = square
root of 2, theta = - pi / 4.
630
00:36:56 --> 00:36:58
And these are both legal.
631
00:36:58 --> 00:37:00
These are perfectly
legal representatives.
632
00:37:00 --> 00:37:03
And that's what I meant
by saying that these
633
00:37:03 --> 00:37:06
representations over here
are somewhat ambiguous.
634
00:37:06 --> 00:37:08
There's more than one answer
to this question, of what
635
00:37:08 --> 00:37:11
the polar representation is.
636
00:37:11 --> 00:37:17
A third possibility, which is
even more dicey but also legal,
637
00:37:17 --> 00:37:21
is r = - square root of 2.
638
00:37:21 --> 00:37:25
Theta = 3 pi / 4.
639
00:37:25 --> 00:37:30
Now, what that corresponds to
doing is going around to here.
640
00:37:30 --> 00:37:33
We're pointing out
3/4 pi, direction.
641
00:37:33 --> 00:37:37
But then going negative
square root of 2, distance.
642
00:37:37 --> 00:37:39
We're going backwards.
643
00:37:39 --> 00:37:42
So we're landing in
the same place.
644
00:37:42 --> 00:37:44
So this is also legal.
645
00:37:44 --> 00:37:44
Yeah.
646
00:37:44 --> 00:37:51
STUDENT: [INAUDIBLE]
647
00:37:51 --> 00:37:53
PROFESSOR: The question is,
don't the radiuses have to be
648
00:37:53 --> 00:37:56
positive because they represent
a distance to the origin?
649
00:37:56 --> 00:38:00
The answer is I
lied to you here.
650
00:38:00 --> 00:38:04
All of these things that I said
are wrong, except for this.
651
00:38:04 --> 00:38:09
Which is the rule for what
polar coordinates mean.
652
00:38:09 --> 00:38:21
So it's maybe plus or minus the
distance, is what it is always.
653
00:38:21 --> 00:38:29
I try not to lie to you too
much, but I do succeed.
654
00:38:29 --> 00:38:36
Now, let's do a little
bit more practice here.
655
00:38:36 --> 00:38:39
There are some easy examples,
which I will run through very
656
00:38:39 --> 00:38:44
quickly. r = a, we already
know this is a circle.
657
00:38:44 --> 00:38:51
And the 3 theta = a
constant is a ray.
658
00:38:51 --> 00:38:55
However, this involves an
implicit assumption, which I
659
00:38:55 --> 00:38:57
want to point out to you.
660
00:38:57 --> 00:38:59
So this is Example 3.
661
00:38:59 --> 00:39:01
Theta's equal to a
constant as a ray.
662
00:39:01 --> 00:39:14
But this implicitly assumes
0 <= r < infinity.
663
00:39:14 --> 00:39:19
If you really wanted to allow
minus infinity < r < infinity
664
00:39:19 --> 00:39:22
in this example, you
would get a line.
665
00:39:22 --> 00:39:28
Gives the whole line.
666
00:39:28 --> 00:39:30
It gives everything behind.
667
00:39:30 --> 00:39:33
So you go out on some ray, you
go backwards on that ray and
668
00:39:33 --> 00:39:36
you get the whole line through
the origin, both ways.
669
00:39:36 --> 00:39:39
If you allow r going to
minus infinity as well.
670
00:39:39 --> 00:39:42
So the typical conventions,
so here are the
671
00:39:42 --> 00:39:49
typical conventions.
672
00:39:49 --> 00:39:53
And you will see people assume
this without even telling you.
673
00:39:53 --> 00:39:55
So you need to
watch out for it.
674
00:39:55 --> 00:39:57
The typical conventions are
certainly this one, which
675
00:39:57 --> 00:40:00
is a nice thing to do.
676
00:40:00 --> 00:40:04
Pretty much all the time,
although not all the time.
677
00:40:04 --> 00:40:05
Most of the time.
678
00:40:05 --> 00:40:12
And then you might have theta
ranging from minus pi to
679
00:40:12 --> 00:40:15
pi, so in other words
symmmetric around 0.
680
00:40:15 --> 00:40:21
Or, another very popular
choice is this one.
681
00:40:21 --> 00:40:25
Theta's >= 0 and strictly
less than 2 pi.
682
00:40:25 --> 00:40:31
So these are the two typical
ranges in which all of these
683
00:40:31 --> 00:40:33
variables are chosen.
684
00:40:33 --> 00:40:34
But not always.
685
00:40:34 --> 00:40:43
You'll find that it's
not consistent.
686
00:40:43 --> 00:40:46
As I said, our job is
to get used to this.
687
00:40:46 --> 00:40:49
And I need to work up
to some slightly more
688
00:40:49 --> 00:40:51
complicated examples.
689
00:40:51 --> 00:40:57
Some of which I'll give
you on next Tuesday.
690
00:40:57 --> 00:41:05
But let's do a few more.
691
00:41:05 --> 00:41:10
So, I guess this is Example 4.
692
00:41:10 --> 00:41:14
Example 4, I'm going
to take y = 1.
693
00:41:14 --> 00:41:20
That's awfully simple in
rectangular coordinates.
694
00:41:20 --> 00:41:24
But interestingly, you might
conceivably want to deal with
695
00:41:24 --> 00:41:26
it in polar coordinates.
696
00:41:26 --> 00:41:29
If you do, so here's how
you make the translation.
697
00:41:29 --> 00:41:32
But this translation
is not so terrible.
698
00:41:32 --> 00:41:39
What you do is, you plug
in y = r sin theta.
699
00:41:39 --> 00:41:40
That's all you have to do.
700
00:41:40 --> 00:41:42
And so that's going
to be equal to 1.
701
00:41:42 --> 00:41:46
And that's going to give
us our polar equation.
702
00:41:46 --> 00:41:50
The polar equation is
r = 1 / sin theta.
703
00:41:50 --> 00:41:54
There it is.
704
00:41:54 --> 00:41:58
And let's draw a picture of it.
705
00:41:58 --> 00:42:03
So here's a picture
of the line y = 1.
706
00:42:03 --> 00:42:09
And now we see that if we
take our rays going out from
707
00:42:09 --> 00:42:17
here, they collide with the
line at various lengths.
708
00:42:17 --> 00:42:20
So if you take an angle, theta,
here there'll be a distance r
709
00:42:20 --> 00:42:22
corresponding to that and
you'll hit this in
710
00:42:22 --> 00:42:23
exactly one spot.
711
00:42:23 --> 00:42:26
For each theta you'll
have a different radius.
712
00:42:26 --> 00:42:27
And it's a variable radius.
713
00:42:27 --> 00:42:30
It's given by this
formula here.
714
00:42:30 --> 00:42:33
And so to trace this line out,
you actually have to realize
715
00:42:33 --> 00:42:36
that there's one more
thing involved.
716
00:42:36 --> 00:42:40
Which is the possible
range of theta.
717
00:42:40 --> 00:42:41
Again, when you're doing
integrations you're going
718
00:42:41 --> 00:42:43
to need to know those
limits of integration.
719
00:42:43 --> 00:42:46
So you're going to
need to know this.
720
00:42:46 --> 00:42:49
The range here goes from theta
= 0, that's sort of when
721
00:42:49 --> 00:42:51
it's out at infinity.
722
00:42:51 --> 00:42:53
That's when the
denominator is 0 here.
723
00:42:53 --> 00:42:55
And it goes all the way to pi.
724
00:42:55 --> 00:42:57
Swing around just
one half-turn.
725
00:42:57 --> 00:43:03
So the range here
is 0 < theta < pi.
726
00:43:03 --> 00:43:04
Yeah, question.
727
00:43:04 --> 00:43:10
STUDENT: [INAUDIBLE]
728
00:43:10 --> 00:43:12
PROFESSOR: The question is, is
it typical to express r as a
729
00:43:12 --> 00:43:16
function of theta, or vice
versa, or does it matter?
730
00:43:16 --> 00:43:20
The answer is that for the
purposes of this course, we're
731
00:43:20 --> 00:43:24
almost always going to be
writing things in this form.
732
00:43:24 --> 00:43:27
r as a function of theta.
733
00:43:27 --> 00:43:30
And you can do
whatever you want.
734
00:43:30 --> 00:43:33
This turns out to be what
we'll be doing in this
735
00:43:33 --> 00:43:37
course, exclusively.
736
00:43:37 --> 00:43:40
As you'll see when we get to
other examples, it's the
737
00:43:40 --> 00:43:43
traditional sort of thing to do
when you're thinking about
738
00:43:43 --> 00:43:48
observing a planet or
something like that.
739
00:43:48 --> 00:43:52
You see the angle, and then
you guess far away it is.
740
00:43:52 --> 00:43:55
But it's not necessary.
741
00:43:55 --> 00:43:58
The formulas are often
easier this way.
742
00:43:58 --> 00:44:00
For the examples that we have.
743
00:44:00 --> 00:44:02
Because it's usually a
trig function of theta.
744
00:44:02 --> 00:44:05
Whereas the other way, it would
be an inverse trig function.
745
00:44:05 --> 00:44:08
So it's an uglier expression.
746
00:44:08 --> 00:44:10
As you can see.
747
00:44:10 --> 00:44:12
The real reason is that we
choose this thing that's
748
00:44:12 --> 00:44:19
easier to deal with.
749
00:44:19 --> 00:44:22
So now let me give you a
slightly more complicated
750
00:44:22 --> 00:44:24
example of the same type.
751
00:44:24 --> 00:44:28
Where we use a shortcut.
752
00:44:28 --> 00:44:31
This is a standard example.
753
00:44:31 --> 00:44:33
And it comes up a lot.
754
00:44:33 --> 00:44:40
And so this is an
off-center circle.
755
00:44:40 --> 00:44:45
A circle is really easy to
describe, but not necessarily
756
00:44:45 --> 00:44:54
if the center is on
the rim of the circle.
757
00:44:54 --> 00:44:56
So that's a different problem.
758
00:44:56 --> 00:44:59
And let's do this with
a circle of radius a.
759
00:44:59 --> 00:45:06
So this is the point (a,
0) and this is (2a, 0).
760
00:45:06 --> 00:45:09
And actually, if you know these
two numbers, you'll be able
761
00:45:09 --> 00:45:11
to remember the result
of this calculation.
762
00:45:11 --> 00:45:13
Which you'll do about five or
six times and then finally
763
00:45:13 --> 00:45:17
you'll memorize it during 18.02
when you will need it a lot.
764
00:45:17 --> 00:45:21
So this is a standard
calculation here.
765
00:45:21 --> 00:45:24
So the starting place is
the rectangular equation.
766
00:45:24 --> 00:45:27
And we're going to pass to
the polar representation.
767
00:45:27 --> 00:45:33
The rectangular representation
is (x - a) ^2 + y ^2 = a ^2.
768
00:45:33 --> 00:45:40
So this is a circle centered
at (a, 0) of radius a.
769
00:45:40 --> 00:45:44
And now, if you like, the slow
way of doing this would be to
770
00:45:44 --> 00:45:50
plug in x = r cos theta,
y = r sin theta.
771
00:45:50 --> 00:45:51
The way I did in
this first step.
772
00:45:51 --> 00:45:53
And that works perfectly well.
773
00:45:53 --> 00:45:56
But I'm going to do it
more quickly than that.
774
00:45:56 --> 00:46:00
Because I can sort of see in
advance how it's going to work.
775
00:46:00 --> 00:46:09
I'm just going to
expand this out.
776
00:46:09 --> 00:46:13
And now I see the
a ^2's cancel.
777
00:46:13 --> 00:46:17
And not only that, but
x^2 + y &2 = r ^2.
778
00:46:17 --> 00:46:19
So this becomes r ^2.
779
00:46:19 --> 00:46:28
That's x ^2 + y ^2 - 2ax = 0.
780
00:46:28 --> 00:46:36
The r came from the fact
that r ^2 = x ^2 + y ^2.
781
00:46:36 --> 00:46:37
So I'm doing this
the rapid way.
782
00:46:37 --> 00:46:42
You can do it by plugging
in, as I said. r equals.
783
00:46:42 --> 00:46:44
So now that I've simplified
it, I am going to
784
00:46:44 --> 00:46:45
use that procedure.
785
00:46:45 --> 00:46:47
I'm going to plug in.
786
00:46:47 --> 00:46:57
So here I have r ^2 -
2a r cos theta = 0.
787
00:46:57 --> 00:47:00
I just plugged in for x.
788
00:47:00 --> 00:47:02
As I said, I could have done
that at the beginning.
789
00:47:02 --> 00:47:06
I just simplified first.
790
00:47:06 --> 00:47:11
And now, this is the same thing
as r ^2 = 2ar cos theta.
791
00:47:11 --> 00:47:13
And we're almost done.
792
00:47:13 --> 00:47:19
There's a boring part of this
equation, which is r = 0.
793
00:47:19 --> 00:47:22
And then there's, if I divide
by r, there's the interesting
794
00:47:22 --> 00:47:23
part of the equation.
795
00:47:23 --> 00:47:25
Which is this.
796
00:47:25 --> 00:47:28
So this is or r = 0.
797
00:47:28 --> 00:47:33
Which is already included
in that equation anyway.
798
00:47:33 --> 00:47:36
So I'm allowed to divide by r
because in the case of r = 0,
799
00:47:36 --> 00:47:39
this is represented anyway.
800
00:47:39 --> 00:47:40
Question.
801
00:47:40 --> 00:47:44
STUDENT: [INAUDIBLE]
802
00:47:44 --> 00:47:46
PROFESSOR: r = 0
is just one case.
803
00:47:46 --> 00:47:48
That is, it's the
union of these two.
804
00:47:48 --> 00:47:49
It's both.
805
00:47:49 --> 00:47:50
Both are possible.
806
00:47:50 --> 00:47:53
So r = 0 is one point on it.
807
00:47:53 --> 00:47:56
And this is all of it.
808
00:47:56 --> 00:48:01
So we can just ignore this.
809
00:48:01 --> 00:48:04
So now I want to say one
more important thing.
810
00:48:04 --> 00:48:06
You need to understand
the range of this.
811
00:48:06 --> 00:48:09
So wait a second and we're
going to figure out
812
00:48:09 --> 00:48:10
the range here.
813
00:48:10 --> 00:48:13
The range is very important,
because otherwise you'll never
814
00:48:13 --> 00:48:18
be able to integrate using
this representation here.
815
00:48:18 --> 00:48:19
So this is the representation.
816
00:48:19 --> 00:48:25
But notice when theta =
0, we're out here at 2a.
817
00:48:25 --> 00:48:27
That's consistent, and that's
actually how you remember
818
00:48:27 --> 00:48:29
this factor 2a here.
819
00:48:29 --> 00:48:30
Because if you remember
this picture and where
820
00:48:30 --> 00:48:34
you land when theta = 0.
821
00:48:34 --> 00:48:36
So that's the theta = 0 part.
822
00:48:36 --> 00:48:40
But now as I tip up like this,
you see that when we get
823
00:48:40 --> 00:48:43
to vertical, we're done.
824
00:48:43 --> 00:48:44
With the circle.
825
00:48:44 --> 00:48:47
It's gotten shorter and shorter
and shorter, and at theta
826
00:48:47 --> 00:48:49
= pi / 2, we're down at 0.
827
00:48:49 --> 00:48:51
Because that's cos pi / 2 = 0.
828
00:48:51 --> 00:48:53
So it swings up like this.
829
00:48:53 --> 00:48:55
And it gets up to pi / 2.
830
00:48:55 --> 00:48:57
Similarly, we swing
down like this.
831
00:48:57 --> 00:48:59
And then we're done.
832
00:48:59 --> 00:49:04
So the range is - pi /
2 < theta < pi / 2.
833
00:49:04 --> 00:49:07
Or, if you want to throw in
the r = 0 case, you can throw
834
00:49:07 --> 00:49:11
in this, this is repeating,
if you like, at the ends.
835
00:49:11 --> 00:49:14
So this is the range
of this circle.
836
00:49:14 --> 00:49:17
And let's see.
837
00:49:17 --> 00:49:21
Next time we'll figure out
area in polar coordinates.
838
00:49:21 --> 00:49:22