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PROFESSOR: So again,
welcome back.
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00:00:24 --> 00:00:28
And today's topic is a
continuation of what
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00:00:28 --> 00:00:29
we did last time.
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00:00:29 --> 00:00:32
We still have a little bit
of work and thinking to do
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00:00:32 --> 00:00:38
concerning polar coordinates.
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00:00:38 --> 00:00:50
So we're going to talk
about polar coordinates.
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And my first job today is to
talk a little bit about area.
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00:00:58 --> 00:01:01
That's something we didn't
mention last time.
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00:01:01 --> 00:01:06
And since we're all back from
Thanksgiving, we can certainly
18
00:01:06 --> 00:01:10
talk about it in
terms of a pie.
19
00:01:10 --> 00:01:14
Which is the basic idea for
area in polar coordinates.
20
00:01:14 --> 00:01:21
Here's our pie, and here's
a slice of the pie.
21
00:01:21 --> 00:01:25
The slice has a piece of arc
length on it, which I'm
22
00:01:25 --> 00:01:28
going to call delta theta.
23
00:01:28 --> 00:01:31
And the area of that
shaded-in slice, I'm
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going to call delta A.
25
00:01:35 --> 00:01:38
And let's suppose that
the radius is a.
26
00:01:38 --> 00:01:39
Little a.
27
00:01:39 --> 00:01:45
So this is a pie of radius a.
28
00:01:45 --> 00:01:48
That's our picture.
29
00:01:48 --> 00:01:51
Now, it's pretty easy to
figure out what the area
30
00:01:51 --> 00:01:54
that slice of pie is.
31
00:01:54 --> 00:01:58
The total area is,
of course, pi a ^2.
32
00:01:58 --> 00:02:00
We know that.
33
00:02:00 --> 00:02:05
And to get this fraction, delta
A, all we have to do is take
34
00:02:05 --> 00:02:10
the percentage of the arc of
the total circumference.
35
00:02:10 --> 00:02:13
That's delta theta / 2 pi.
36
00:02:13 --> 00:02:18
This is the fraction of area --
sorry, fraction of the total
37
00:02:18 --> 00:02:20
circumference, the total
length around the rim.
38
00:02:20 --> 00:02:24
And then we multiply
that by pi a ^2.
39
00:02:24 --> 00:02:27
And that's giving
us the total area.
40
00:02:27 --> 00:02:30
And if you work that out,
that's delta A is equal to,
41
00:02:30 --> 00:02:36
the pi's cancel and we
have 1/2 a ^2 delta theta.
42
00:02:36 --> 00:02:44
So here's the basic formula.
43
00:02:44 --> 00:02:49
And now what we need to
do is to talk about
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00:02:49 --> 00:02:54
a variable pie here.
45
00:02:54 --> 00:02:58
That would be a pie with
a kind of a wavy crust.
46
00:02:58 --> 00:03:01
Which is coming
around like this.
47
00:03:01 --> 00:03:04
So r = r (theta).
48
00:03:04 --> 00:03:10
The distance from the center is
varying with the place where we
49
00:03:10 --> 00:03:13
are, the angle where
we're shooting out.
50
00:03:13 --> 00:03:22
And now I want to subdivide
that into little chunks here.
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00:03:22 --> 00:03:26
Now, the idea for adding up the
area, the total area of this
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00:03:26 --> 00:03:31
piece that's swept out, is to
break it up into little slices
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whose areas are almost
easy to calculate.
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00:03:37 --> 00:03:43
Namely, what we're going
to do is to take, and I'm
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going to label it this way.
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00:03:46 --> 00:03:49
I'm going to take these little
circular arcs, which go.
57
00:03:49 --> 00:03:55
So I'm going to extend
past where this goes.
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00:03:55 --> 00:03:58
And then I'm going to take
each circular arc here.
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So here's a circular arc.
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00:04:00 --> 00:04:04
And then here's
another circular arc.
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And here's another
circular arc.
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It's just right on the
nose in that case.
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Now, in these two cases, so
basically the picture that I'm
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trying to draw for you is this.
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I have some sector.
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And then I have
some circular arc.
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And maybe it takes
a little extra.
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There's a little extra
area, I'm making an
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error in the area.
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This is a little extra area.
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And maybe to draw
it the other way.
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I'm a little short on this one.
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And let's say on this one
I'm right on the nose.
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I have the same arc as the
curve of the surface.
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Now this is a little bit like
the step functions that
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we used in Riemann sums.
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It's practically the same.
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Eventually, this little band of
stuff that we're missing by, if
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we take very, very narrow
little slices here, is
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going to be negligible.
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00:04:58 --> 00:05:01
It'll get closer and closer
to the curve itself.
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00:05:01 --> 00:05:04
So that area will tend
to 0 in the limit.
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00:05:04 --> 00:05:05
So we don't have to
worry about it.
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00:05:05 --> 00:05:09
And the approximate
relationship is sitting here.
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Where this distance now is r.
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00:05:12 --> 00:05:14
So this radius is r.
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00:05:14 --> 00:05:18
And this is this delta theta.
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00:05:18 --> 00:05:23
And so in the approximate case,
what we have is that delta A is
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approximately 1/2
r^2 delta theta.
90
00:05:28 --> 00:05:30
Which is practically the
same thing we had here.
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00:05:30 --> 00:05:34
Except that that r is
replacing the constant there.
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00:05:34 --> 00:05:39
And it's approximately true,
because r is varying.
93
00:05:39 --> 00:05:42
And then in the limit, we
have the exact formula
94
00:05:42 --> 00:05:43
for the differential.
95
00:05:43 --> 00:05:46
Which is this one.
96
00:05:46 --> 00:05:49
So this is the main
formula for area.
97
00:05:49 --> 00:05:52
And if you like, the total
area then is going to be the
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00:05:52 --> 00:05:55
integral from some starting
place to some end place
99
00:05:55 --> 00:06:00
of 1/2 r ^2 d theta.
100
00:06:00 --> 00:06:04
Now, this is only useful in
the situation that we're in.
101
00:06:04 --> 00:06:07
Namely, so this is the
other important formula.
102
00:06:07 --> 00:06:11
And this is only useful when
r is a function of theta.
103
00:06:11 --> 00:06:17
When this is the way in which
the region is presented to us.
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00:06:17 --> 00:06:20
So that's the setup.
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00:06:20 --> 00:06:23
And that's our main formula.
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Let's do what example.
107
00:06:28 --> 00:06:30
The example that I'm going to
take is the one that we did
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00:06:30 --> 00:06:33
at the end of last time, or
near the end of last time.
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00:06:33 --> 00:06:39
Which was this formula
here. r = 2a cos theta.
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00:06:39 --> 00:06:46
Remember, that was the same
as (x - a) ^2 + y ^2 = a ^2.
111
00:06:46 --> 00:06:49
So this is what we
did last time.
112
00:06:49 --> 00:06:54
We connected this rectangular
representation to that
113
00:06:54 --> 00:06:54
polar representation.
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00:06:54 --> 00:07:02
And the picture is of a circle.
115
00:07:02 --> 00:07:11
Where this is the
point (2a, 0).
116
00:07:11 --> 00:07:16
So let's figure out
what the area is.
117
00:07:16 --> 00:07:21
Well, first of all, we have to
figure out when we sweep out
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00:07:21 --> 00:07:23
the area, we have to realize
that we only go from
119
00:07:23 --> 00:07:27
- pi / 2 to pi / 2.
120
00:07:27 --> 00:07:30
So that's something we can
get from the picture.
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00:07:30 --> 00:07:33
You can also get it directly
from this formula if you
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00:07:33 --> 00:07:38
realize that cosine is
positive in this range here.
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And at the ends, it's 0.
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00:07:40 --> 00:07:46
So the thing encloses a
region at these ends.
125
00:07:46 --> 00:07:54
So at the ends, cosine of
plus or minus pi / 2 = 0.
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00:07:54 --> 00:08:02
That's what synchs this up like
a little sack, if you like.
127
00:08:02 --> 00:08:06
So the area is now going to be
the integral from - pi / 2 to
128
00:08:06 --> 00:08:11
pi / 2 of 1/2 times the square
of r, that's (2a cos
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00:08:11 --> 00:08:18
theta)^2 d theta.
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00:08:18 --> 00:08:18
Question.
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00:08:18 --> 00:08:25
STUDENT: [INAUDIBLE]
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PROFESSOR: How do I know from
looking at the picture that
133
00:08:27 --> 00:08:33
I'm going from - pi / 2 to
pi / 2, is the question.
134
00:08:33 --> 00:08:36
I do it with my whole body.
135
00:08:36 --> 00:08:39
I say, here I am pointing down.
136
00:08:39 --> 00:08:41
That's - pi / 2.
137
00:08:41 --> 00:08:43
I sweep up, that's 0.
138
00:08:43 --> 00:08:47
And I get all the way up
to here. that's pi / 2.
139
00:08:47 --> 00:08:48
So that's the way I do it.
140
00:08:48 --> 00:08:51
That's really the way I
do it, I'm being honest.
141
00:08:51 --> 00:08:54
Now if you're a machine,
you can't actually look.
142
00:08:54 --> 00:08:57
And you don't have a body, so
you can't point your arms.
143
00:08:57 --> 00:08:59
Then you would have to
go by the formulas.
144
00:08:59 --> 00:09:02
And you'd have to actually
use something like
145
00:09:02 --> 00:09:03
this formula here.
146
00:09:03 --> 00:09:07
The fact that this is
where the loop syncs up.
147
00:09:07 --> 00:09:10
This is where the
radius comes into 0.
148
00:09:10 --> 00:09:11
At pi / 2.
149
00:09:11 --> 00:09:17
So you need to know that in
order to understand the range.
150
00:09:17 --> 00:09:17
Another question.
151
00:09:17 --> 00:09:23
STUDENT: [INAUDIBLE]
152
00:09:23 --> 00:09:26
PROFESSOR: So when we're doing
each, and we just guess that
153
00:09:26 --> 00:09:28
it's going to be a loop.
154
00:09:28 --> 00:09:30
I'm probably going to
give you some clues as
155
00:09:30 --> 00:09:31
to what's going on.
156
00:09:31 --> 00:09:33
Because it's very hard to
figure these things out.
157
00:09:33 --> 00:09:37
Sometimes it'll be bounded by
one curb and another curb, and
158
00:09:37 --> 00:09:39
I'll say it's the thing in
between those two curbs.
159
00:09:39 --> 00:09:42
That's the kind of
thing that I could do.
160
00:09:42 --> 00:09:47
Here, you really should
know this one in advance.
161
00:09:47 --> 00:09:51
This is by far the most,
or this is one of the
162
00:09:51 --> 00:09:52
typical cases, anyway.
163
00:09:52 --> 00:09:54
I'm going to give you a
couple more examples.
164
00:09:54 --> 00:09:57
Don't get too worked
up over this.
165
00:09:57 --> 00:09:59
You will somehow be
able to visualize it.
166
00:09:59 --> 00:10:05
I'll give you some examples to
help you out with it later.
167
00:10:05 --> 00:10:06
So here's the situation.
168
00:10:06 --> 00:10:07
Here's my integral.
169
00:10:07 --> 00:10:10
And now we're faced
with a trig integral.
170
00:10:10 --> 00:10:12
Which we have to
remember how to do.
171
00:10:12 --> 00:10:15
Now, the trig integral here
-- so first let me factor
172
00:10:15 --> 00:10:16
out the constants.
173
00:10:16 --> 00:10:22
This is 4a ^2 / 2, so it's 2a
^2 integral from - pi / 2 to pi
174
00:10:22 --> 00:10:26
/ 2 of cos ^2 theta d theta.
175
00:10:26 --> 00:10:29
And now you have to remember
what you're supposed
176
00:10:29 --> 00:10:32
to do at this point.
177
00:10:32 --> 00:10:35
So think, if you haven't done
it yet, this is practice
178
00:10:35 --> 00:10:37
you need to do.
179
00:10:37 --> 00:10:40
This trig integral is handled
by a double angle formula.
180
00:10:40 --> 00:10:43
As it happens, I'm going to
be giving you these formulas
181
00:10:43 --> 00:10:44
on the review sheet.
182
00:10:44 --> 00:10:47
You'll see they're written
on the review sheet.
183
00:10:47 --> 00:10:49
At least in some form.
184
00:10:49 --> 00:10:52
So for example, there's a
formula, and this will
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00:10:52 --> 00:10:54
be on the exam, too.
186
00:10:54 --> 00:10:58
So this is the correct
formula to use here.
187
00:10:58 --> 00:11:04
Is that this is 1 + cos
2 theta / 2 d theta.
188
00:11:04 --> 00:11:08
So that's the substitution that
you use for the cosine ^2
189
00:11:08 --> 00:11:12
in order to integrate it.
190
00:11:12 --> 00:11:15
That serves as a little
review of trig integrals.
191
00:11:15 --> 00:11:18
And now, this is quite easy.
192
00:11:18 --> 00:11:27
This integral now is easy.
193
00:11:27 --> 00:11:28
Why is it easy?
194
00:11:28 --> 00:11:31
Well, because it's the
antiderivative of a constant,
195
00:11:31 --> 00:11:33
and cos 2 theta its
antiderivative you're supposed
196
00:11:33 --> 00:11:34
to be able to write down.
197
00:11:34 --> 00:11:36
So the antiderivative
of 1 is theta.
198
00:11:36 --> 00:11:41
And the antiderivative
of the cos is 1/2 sin
199
00:11:41 --> 00:11:50
when it's 2 theta.
200
00:11:50 --> 00:11:55
And that is a ^2 (pi
/2 - (- pi) / 2).
201
00:11:55 --> 00:12:00
And the signs go away
because they're both 0.
202
00:12:00 --> 00:12:04
So all told we get pi a ^2,
which is certainly what
203
00:12:04 --> 00:12:05
we would like it to be.
204
00:12:05 --> 00:12:09
It's the area of the circle.
205
00:12:09 --> 00:12:10
Another question?
206
00:12:10 --> 00:12:29
STUDENT: [INAUDIBLE]
207
00:12:29 --> 00:12:30
PROFESSOR: The question,
so I'm not sure which
208
00:12:30 --> 00:12:31
question you're asking.
209
00:12:31 --> 00:12:35
I pivoted my arm around (0, 0).
210
00:12:35 --> 00:12:37
This point, this is the
point we're talking about,
211
00:12:37 --> 00:12:39
(0, 0), is a key point.
212
00:12:39 --> 00:12:43
It's where I guess you could
say I stuck my elbow there.
213
00:12:43 --> 00:12:48
Now, the reason is that it's
the place where r = 0.
214
00:12:48 --> 00:12:50
So it's more or less the center
of the universe from the point
215
00:12:50 --> 00:12:54
of view of this problem.
216
00:12:54 --> 00:12:58
So it's the reference point and
if you like, when you're doing
217
00:12:58 --> 00:13:01
this, it's a little bit
like a radar screen.
218
00:13:01 --> 00:13:03
Everything is centered at the
origin and you're taking
219
00:13:03 --> 00:13:07
rays coming out from it.
220
00:13:07 --> 00:13:10
And seeing where
they're going to go.
221
00:13:10 --> 00:13:12
So for example, this is the
theta = 0 ray, this is
222
00:13:12 --> 00:13:15
the theta = pi / 4 ray.
223
00:13:15 --> 00:13:17
This the theta = pi / 2 ray.
224
00:13:17 --> 00:13:21
And indeed, if my elbow is
right at this center here,
225
00:13:21 --> 00:13:24
I'm pointing in those
various directions.
226
00:13:24 --> 00:13:32
So that's what I had in
mind when I did that.
227
00:13:32 --> 00:13:35
You can always get these
formulas, by the way, from
228
00:13:35 --> 00:13:42
the original business, x = r
cos theta, y = r sin theta.
229
00:13:42 --> 00:13:45
But it's useful to have the
geometric picture as well.
230
00:13:45 --> 00:13:46
In other words, if you were
a machine you'd have to
231
00:13:46 --> 00:13:48
rely on these formulas.
232
00:13:48 --> 00:13:49
And plot things using these.
233
00:13:49 --> 00:13:57
Always.
234
00:13:57 --> 00:14:00
Now, in terms of plotting
I want to expand your
235
00:14:00 --> 00:14:01
brain a little bit.
236
00:14:01 --> 00:14:03
So we need just a little bit
more practice with plotting.
237
00:14:03 --> 00:14:05
In polar coordinates.
238
00:14:05 --> 00:14:10
And so, the first question
that I want to ask you is,
239
00:14:10 --> 00:14:15
what happens outside of
this range of theta?
240
00:14:15 --> 00:14:19
In other words, what happens
if theta's beyond pi / 2?
241
00:14:19 --> 00:14:22
Can somebody see what's
happening to the
242
00:14:22 --> 00:14:23
formulas in that case?
243
00:14:23 --> 00:14:27
So what I'm looking at
now, let's go back to it.
244
00:14:27 --> 00:14:35
What I'm looking at is
this formula here.
245
00:14:35 --> 00:14:38
But to use the elbow
analogy here, I'm swept
246
00:14:38 --> 00:14:39
around like this.
247
00:14:39 --> 00:14:40
But now I'm going
to point this way.
248
00:14:40 --> 00:14:42
I'm going to point
out over there.
249
00:14:42 --> 00:14:49
My hand is up here in the
northwest direction.
250
00:14:49 --> 00:14:51
So what's going to happen?
251
00:14:51 --> 00:14:52
Somebody want to tell me?
252
00:14:52 --> 00:14:55
STUDENT: [INAUDIBLE]
253
00:14:55 --> 00:14:56
PROFESSOR: It goes
around itself.
254
00:14:56 --> 00:14:57
That's right.
255
00:14:57 --> 00:15:02
What happens is that when r
crosses this vertical, r =
256
00:15:02 --> 00:15:05
0, when it crosses over
here it goes negative.
257
00:15:05 --> 00:15:08
So although my theta is
pointing me this way, the thing
258
00:15:08 --> 00:15:10
is going to go backwards.
259
00:15:10 --> 00:15:12
And there's another clue.
260
00:15:12 --> 00:15:13
Which is very important.
261
00:15:13 --> 00:15:15
How far backwards is it going?
262
00:15:15 --> 00:15:17
Well, you don't actually need
to know anything but this
263
00:15:17 --> 00:15:21
equation here, to understand
that it has to be on
264
00:15:21 --> 00:15:23
the same circle.
265
00:15:23 --> 00:15:26
So when I'm pointing this way,
the things points backwards
266
00:15:26 --> 00:15:29
to this point over there.
267
00:15:29 --> 00:15:32
So what happens is,
it goes around once.
268
00:15:32 --> 00:15:34
And then when I point
out this way, it sweeps
269
00:15:34 --> 00:15:35
around a second time.
270
00:15:35 --> 00:15:37
It just keeps on going
around the same circle.
271
00:15:37 --> 00:15:39
So over here it's empty.
272
00:15:39 --> 00:15:41
Because it's pointing the
other way and it's sweeping
273
00:15:41 --> 00:15:42
around the same curve.
274
00:15:42 --> 00:15:47
A second time.
275
00:15:47 --> 00:15:51
Now, if you were foolish enough
to integrate, say, from 0 to 2
276
00:15:51 --> 00:15:54
pi or some wider range, what
would happen is you would
277
00:15:54 --> 00:15:55
just double the area.
278
00:15:55 --> 00:16:00
Because you would have
swept it out twice.
279
00:16:00 --> 00:16:02
So that's the mistake
that you'll make.
280
00:16:02 --> 00:16:05
Sometimes you'll count things
as negative and positive.
281
00:16:05 --> 00:16:07
But because there's a
square here, it's always
282
00:16:07 --> 00:16:08
a positive quantity.
283
00:16:08 --> 00:16:13
And you'll always over-count
if you go too far.
284
00:16:13 --> 00:16:15
So that's what happens.
285
00:16:15 --> 00:16:17
Again, it sweeps out
the same region.
286
00:16:17 --> 00:16:20
That's because these two
equations really are
287
00:16:20 --> 00:16:22
equivalent to each other.
288
00:16:22 --> 00:16:24
It's just that this one
sweeps it out twice.
289
00:16:24 --> 00:16:28
And this one doesn't say
how it's sweeping it out.
290
00:16:28 --> 00:16:30
Yeah, another question.
291
00:16:30 --> 00:16:32
STUDENT: Doesn't this equation
also work if you just
292
00:16:32 --> 00:16:34
go from 0 to pi?
293
00:16:34 --> 00:16:36
PROFESSOR: Does the
integration work if you
294
00:16:36 --> 00:16:38
just go from 0 to pi?
295
00:16:38 --> 00:16:40
The answer is yes.
296
00:16:40 --> 00:16:43
That's a very weird
object, though.
297
00:16:43 --> 00:16:44
Let me just show
you what that is.
298
00:16:44 --> 00:16:47
If you started from 0 to 2 pi.
299
00:16:47 --> 00:16:50
So I'll illustrate it on here.
300
00:16:50 --> 00:16:53
The first thing that you swept
out between 0 and pi over
301
00:16:53 --> 00:16:54
2 is this part here.
302
00:16:54 --> 00:16:56
That was swept out.
303
00:16:56 --> 00:17:00
And then, when you're going
around this next quadrant here,
304
00:17:00 --> 00:17:05
you're actually sweeping
out this underside here.
305
00:17:05 --> 00:17:07
So actually, you're getting it
because you're getting half of
306
00:17:07 --> 00:17:09
it on one half, and getting the
other half on the
307
00:17:09 --> 00:17:10
other quadrant.
308
00:17:10 --> 00:17:14
So it's actually giving
you the right answer.
309
00:17:14 --> 00:17:15
That turns out to be OK.
310
00:17:15 --> 00:17:17
It's a little weird way
to chop up a circle.
311
00:17:17 --> 00:17:23
But it's legal.
312
00:17:23 --> 00:17:25
But of course, that's
an accident of this
313
00:17:25 --> 00:17:26
particular figure.
314
00:17:26 --> 00:17:27
You can't count on
that happening.
315
00:17:27 --> 00:17:29
It's much better to line
it up exactly with
316
00:17:29 --> 00:17:32
what the figure does.
317
00:17:32 --> 00:17:35
So don't do that too often.
318
00:17:35 --> 00:17:40
You might run into troubles.
319
00:17:40 --> 00:17:44
So I'm going to give you a
couple more examples of
320
00:17:44 --> 00:17:48
practice with these pictures.
321
00:17:48 --> 00:17:57
And maybe I'm going to get
rid of this one up here.
322
00:17:57 --> 00:18:03
So here's another favorite.
323
00:18:03 --> 00:18:05
Here's another favorite.
324
00:18:05 --> 00:18:08
So this, if you
like, is Example 2.
325
00:18:08 --> 00:18:09
I guess we had an
Example 1 up there.
326
00:18:09 --> 00:18:12
And now we're really not
going to try to do any
327
00:18:12 --> 00:18:13
more area examples.
328
00:18:13 --> 00:18:15
The area examples are
actually straightforward.
329
00:18:15 --> 00:18:18
It's really just figuring out
what the picture looks like.
330
00:18:18 --> 00:18:27
So this is examples
of drawings.
331
00:18:27 --> 00:18:34
So this one is one that's
kind of fun to do.
332
00:18:34 --> 00:18:37
This is r = sin 2 theta.
333
00:18:37 --> 00:18:40
Something like this
is on your homework.
334
00:18:40 --> 00:18:46
And so what happens
here is the following.
335
00:18:46 --> 00:18:51
What happens here is that
at theta = 0, that's
336
00:18:51 --> 00:18:53
the first place.
337
00:18:53 --> 00:18:56
So let's just plot
a few places here.
338
00:18:56 --> 00:18:57
I'm not going to
plot very many.
339
00:18:57 --> 00:19:00
Theta's = 0, I get r as 1.
340
00:19:00 --> 00:19:02
Whoops, I get r is 0.
341
00:19:02 --> 00:19:04
Sorry.
342
00:19:04 --> 00:19:09
And then pi / 4, that's where I
get sin pi / 2, I get 1 here.
343
00:19:09 --> 00:19:10
For this.
344
00:19:10 --> 00:19:14
And then again, at pi /
2, I get sin pi, which
345
00:19:14 --> 00:19:17
is back at 0 again.
346
00:19:17 --> 00:19:20
So and the other thing
to say is in between
347
00:19:20 --> 00:19:21
here it's positive.
348
00:19:21 --> 00:19:22
In between.
349
00:19:22 --> 00:19:27
So what it does is, it starts
out at 0 and it goes out
350
00:19:27 --> 00:19:30
to the radius 1 over here.
351
00:19:30 --> 00:19:32
And then it comes back.
352
00:19:32 --> 00:19:35
So it does something like this.
353
00:19:35 --> 00:19:39
It goes out, and it comes back.
354
00:19:39 --> 00:19:45
Now because of the symmetries
of the sine function, this
355
00:19:45 --> 00:19:47
is pretty much all
you need to know.
356
00:19:47 --> 00:19:51
It does something similar
in all of the quadrants.
357
00:19:51 --> 00:19:55
But in order to see what
it's doing, it's useful for
358
00:19:55 --> 00:19:57
you to watch me draw it.
359
00:19:57 --> 00:20:00
Because the order is very
important for understanding
360
00:20:00 --> 00:20:01
what it's doing.
361
00:20:01 --> 00:20:06
It's similar to this weird
business with the circle here.
362
00:20:06 --> 00:20:09
So watch me draw this guy.
363
00:20:09 --> 00:20:11
I'll draw it in red because
it usually has a name.
364
00:20:11 --> 00:20:12
So here it is.
365
00:20:12 --> 00:20:15
It does this things.
366
00:20:15 --> 00:20:17
And then it does this.
367
00:20:17 --> 00:20:19
And then it does this.
368
00:20:19 --> 00:20:21
And then it does that.
369
00:20:21 --> 00:20:26
So it's called a
four-leaf rose.
370
00:20:26 --> 00:20:30
I drew it in pink because
it's kind of a rose here.
371
00:20:30 --> 00:20:32
So it started out over here.
372
00:20:32 --> 00:20:34
This is Step 1.
373
00:20:34 --> 00:20:40
And this is the range
0 < theta < pi / 4.
374
00:20:40 --> 00:20:42
It did this part here.
375
00:20:42 --> 00:20:46
And then it went to here.
376
00:20:46 --> 00:20:48
So I should draw these in
white, because they're
377
00:20:48 --> 00:20:50
harder to read in red.
378
00:20:50 --> 00:20:52
But now look at what it did.
379
00:20:52 --> 00:20:55
It did not make a
right angle turn.
380
00:20:55 --> 00:20:56
It was nice and smooth.
381
00:20:56 --> 00:20:59
It went around here and
then it went down here.
382
00:20:59 --> 00:21:00
This is 3.
383
00:21:00 --> 00:21:02
Back here, that's 4.
384
00:21:02 --> 00:21:05
And then over here, that's 5.
385
00:21:05 --> 00:21:07
Back up here, that's 6.
386
00:21:07 --> 00:21:09
And then around here, that's 7.
387
00:21:09 --> 00:21:11
And down here, that's 8.
388
00:21:11 --> 00:21:14
And then back where it started
and goes around again.
389
00:21:14 --> 00:21:18
And this is because actually
it's switching sign when
390
00:21:18 --> 00:21:19
it crosses the origin.
391
00:21:19 --> 00:21:21
When it was over in this
quadrant the first time, it
392
00:21:21 --> 00:21:28
actually was tracing what's
directly behind it.
393
00:21:28 --> 00:21:29
So this is kind of amusing.
394
00:21:29 --> 00:21:32
From this little tiny
formula you get this
395
00:21:32 --> 00:21:34
pretty diagram here.
396
00:21:34 --> 00:21:36
Anyway that's, as I
say, an old favorite.
397
00:21:36 --> 00:21:41
And here if you want to do the
area of one leaf, you've got to
398
00:21:41 --> 00:21:43
make sure you understand that
it's a small piece
399
00:21:43 --> 00:21:49
of the whole.
400
00:21:49 --> 00:21:52
OK, now I have one last
drawing example that I
401
00:21:52 --> 00:21:54
want to discuss with you.
402
00:21:54 --> 00:21:57
And it involves another skill
that we haven't quite gotten
403
00:21:57 --> 00:21:59
enough practice with.
404
00:21:59 --> 00:22:01
So I'm going to do that one.
405
00:22:01 --> 00:22:04
And it's also preparation
for an exercise.
406
00:22:04 --> 00:22:08
But one that we're going
to do after the test.
407
00:22:08 --> 00:22:15
So here's my last example.
408
00:22:15 --> 00:22:17
We're going to discuss
what happens with
409
00:22:17 --> 00:22:20
this function here.
410
00:22:20 --> 00:22:23
Sorry, that's not
legible, is it.
411
00:22:23 --> 00:22:35
That's a cosine. r =
1 / 1 + 2 cos theta.
412
00:22:35 --> 00:22:39
Now, the first thing I want to
do is just take our time a
413
00:22:39 --> 00:22:44
little bit and plot
a few points.
414
00:22:44 --> 00:22:48
So here's the values of theta
and here are the values of r,
415
00:22:48 --> 00:22:49
and we'll see what happens.
416
00:22:49 --> 00:22:52
And we'll try to figure
out what it's doing.
417
00:22:52 --> 00:22:56
When theta = 0, cos = 1.
418
00:22:56 --> 00:23:01
So r = 1/3.
419
00:23:01 --> 00:23:07
The denominator is 1
+ 2, so it's 1/3.
420
00:23:07 --> 00:23:10
If theta, I'm going to
make it easy, we're not
421
00:23:10 --> 00:23:11
going to do so many.
422
00:23:11 --> 00:23:16
I'm going to do pi / 2, that's
an easy value of the cosine.
423
00:23:16 --> 00:23:18
That's cos pi / 2 = 0.
424
00:23:18 --> 00:23:23
So that value of r = 1.
425
00:23:23 --> 00:23:30
And now I'm going to back
up and do - pi / 2. - pi
426
00:23:30 --> 00:23:33
/ 2, again, cosine = 0.
427
00:23:33 --> 00:23:39
And r = 1.
428
00:23:39 --> 00:23:42
So now I'd like to just plot
those points anyway, and
429
00:23:42 --> 00:23:47
see what's going on with
this expression here.
430
00:23:47 --> 00:23:49
The first one is a rectangular.
431
00:23:49 --> 00:23:53
I'm going to write the
rectangular coordinates here,
432
00:23:53 --> 00:23:56
not the polar coordinates.
433
00:23:56 --> 00:24:01
The rectangular coordinates
here 1/3 out at the
434
00:24:01 --> 00:24:04
horizontal, so it's (1/3, 0).
435
00:24:04 --> 00:24:08
The polar coordinates is (1/3,
0), but the rectangular
436
00:24:08 --> 00:24:10
coordinate is also that.
437
00:24:10 --> 00:24:14
And over here, at pi /
2, the distance is 1.
438
00:24:14 --> 00:24:18
So this is the point (0,
1) in x-y coordinates.
439
00:24:18 --> 00:24:26
And then down here at, -
pi / 2, it's (0, - 1).
440
00:24:26 --> 00:24:28
Let me just emphasize.
441
00:24:28 --> 00:24:31
You should be able to think of
this visually if you can crank
442
00:24:31 --> 00:24:33
your arm around and think it.
443
00:24:33 --> 00:24:37
Or if you're right-handed
you'll bend that way now.
444
00:24:37 --> 00:24:38
Anyway.
445
00:24:38 --> 00:24:38
Or you'll have to use.
446
00:24:38 --> 00:24:44
But this also works using
this formulas x = r cos
447
00:24:44 --> 00:24:48
theta, y = r sin theta.
448
00:24:48 --> 00:24:53
Notice that in this case, r
was 1 but the cosine was 0.
449
00:24:53 --> 00:24:57
So you plug in
theta = - pi / 2.
450
00:24:57 --> 00:24:58
And r = 1.
451
00:24:58 --> 00:25:01
And lo and behold,
you get 0 here.
452
00:25:01 --> 00:25:03
And here you get -
1 here you get 1.
453
00:25:03 --> 00:25:05
So this is - 1.
454
00:25:05 --> 00:25:07
So this is an example.
455
00:25:07 --> 00:25:11
I did it purely visually
or sort of organically.
456
00:25:11 --> 00:25:16
But you can also do it by
plugging in the numbers.
457
00:25:16 --> 00:25:21
Now in between, the
denominator is positive.
458
00:25:21 --> 00:25:22
And it's something in between.
459
00:25:22 --> 00:25:26
It's going to sweep around
something like this.
460
00:25:26 --> 00:25:29
That's what happens in between.
461
00:25:29 --> 00:25:34
As theta increases from
- pi / 2 to pi / 2.
462
00:25:34 --> 00:25:37
And now something interesting
happens with this particular
463
00:25:37 --> 00:25:40
function, which is that we
notice that the denominator
464
00:25:40 --> 00:25:43
is 0 at a certain place.
465
00:25:43 --> 00:25:49
Namely, if I solve 2 cos theta
= - 1, then the denominator
466
00:25:49 --> 00:25:51
is going to be 0 there.
467
00:25:51 --> 00:25:58
That's cos theta = - 1/2, so
theta is equal to, it turns
468
00:25:58 --> 00:26:01
out, plus or minus 2 pi / 3.
469
00:26:01 --> 00:26:03
Those are the values here.
470
00:26:03 --> 00:26:07
So when we're out here
somewhere, in these
471
00:26:07 --> 00:26:10
directions, there's nothing.
472
00:26:10 --> 00:26:14
It's going infinitely far out.
473
00:26:14 --> 00:26:20
Those ways.
474
00:26:20 --> 00:26:23
OK that's about as much as
we'll be able to figure out
475
00:26:23 --> 00:26:26
of this diagram without
doing some analytic work.
476
00:26:26 --> 00:26:31
And that's the other little
piece that I want to explain.
477
00:26:31 --> 00:26:34
Namely, going backwards
from polar coordinates to
478
00:26:34 --> 00:26:36
rectangular coordinates.
479
00:26:36 --> 00:26:38
Which is one thing
that we haven't done.
480
00:26:38 --> 00:26:40
So let's do that.
481
00:26:40 --> 00:26:48
So what is the
rectangular equation?
482
00:26:48 --> 00:26:58
That means the (x, y)
equation for this r =
483
00:26:58 --> 00:27:03
1 / 1 + 2 cos theta.
484
00:27:03 --> 00:27:06
And let's see what it is.
485
00:27:06 --> 00:27:08
Well, first I'm going to
clear the denominator here.
486
00:27:08 --> 00:27:15
This is r + 2r cos theta = 1.
487
00:27:15 --> 00:27:23
And now I'm going to rewrite
it as r = 1 - 2r cos theta.
488
00:27:23 --> 00:27:25
And the reason for that
is that in a minute I'll
489
00:27:25 --> 00:27:27
explain to you why.
490
00:27:27 --> 00:27:29
This is 1 - 2x.
491
00:27:29 --> 00:27:32
And this guy, I'm
going to square now.
492
00:27:32 --> 00:27:38
I'm going to make this
r ^2 = (1 - 2x) ^2.
493
00:27:38 --> 00:27:49
And now, with an r^2, I
can plug in x ^2 + y ^2.
494
00:27:49 --> 00:27:53
So this is a standard
thing to do.
495
00:27:53 --> 00:27:55
And it's basically what
you're going to do any
496
00:27:55 --> 00:27:57
time you're faced with
an equation like this.
497
00:27:57 --> 00:27:59
Is try to work it out.
498
00:27:59 --> 00:28:05
And, in these situations where
you have 1 / a + b cos theta,
499
00:28:05 --> 00:28:08
or sin theta, you'll always
come out with some quadratic
500
00:28:08 --> 00:28:12
expression like this.
501
00:28:12 --> 00:28:14
Now, I'm going to
combine terms.
502
00:28:14 --> 00:28:18
So here I have - 3x ^2 +
y ^2, and put everything
503
00:28:18 --> 00:28:20
on the the left side.
504
00:28:20 --> 00:28:24
So that's this.
505
00:28:24 --> 00:28:28
And we recognize, well you're
supposed to recognize, that
506
00:28:28 --> 00:28:36
this is what's known
as a hyperbola.
507
00:28:36 --> 00:28:39
If the signs are the
same, it's an ellipse.
508
00:28:39 --> 00:28:42
If the the signs are
opposite it's a hyperbola.
509
00:28:42 --> 00:28:44
And in between, if one of the
coefficients on the quadratic
510
00:28:44 --> 00:28:49
is 0, it's a parabola.
511
00:28:49 --> 00:28:54
So now we see that the picture
that we drew there is actually,
512
00:28:54 --> 00:28:56
turns out it's going to have
asymptotes, it's going
513
00:28:56 --> 00:29:01
to be a hyperbola.
514
00:29:01 --> 00:29:06
So now, let me ask you the last
little mind-bending question
515
00:29:06 --> 00:29:08
that I want to ask.
516
00:29:08 --> 00:29:10
Which is, what happens.
517
00:29:10 --> 00:29:12
So now I'm using my
right arm, I guess.
518
00:29:12 --> 00:29:14
But my elbow's at
the origin here.
519
00:29:14 --> 00:29:18
What happens if I pass outside,
to the range where this
520
00:29:18 --> 00:29:20
denominator is negative.
521
00:29:20 --> 00:29:24
It crossed 0 and it
went to negative.
522
00:29:24 --> 00:29:28
It's sweeping out
something over here.
523
00:29:28 --> 00:29:32
Is it sweeping out
the same curve?
524
00:29:32 --> 00:29:34
Anybody have any idea
what it's doing?
525
00:29:34 --> 00:29:34
Yeah.
526
00:29:34 --> 00:29:37
STUDENT: [INAUDIBLE]
527
00:29:37 --> 00:29:38
PROFESSOR: Yeah, exactly.
528
00:29:38 --> 00:29:38
Good answer.
529
00:29:38 --> 00:29:44
It's the other branch
of the hyperbola.
530
00:29:44 --> 00:29:46
So what's actually happening is
in disguise, there's another
531
00:29:46 --> 00:29:48
branch of the hyperbola which
is being swept up by the
532
00:29:48 --> 00:29:50
other piece of this thing.
533
00:29:50 --> 00:29:54
Now, that is consistent with
these algebraic equations.
534
00:29:54 --> 00:29:57
The algebraic equation that
I got here doesn't say
535
00:29:57 --> 00:30:00
which branch of the
hyperbola I've got.
536
00:30:00 --> 00:30:05
It's actually got two branches.
537
00:30:05 --> 00:30:08
And the curve really was,
in disguise, capturing
538
00:30:08 --> 00:30:12
both of them.
539
00:30:12 --> 00:30:15
I want to make the connection
now with the basic
540
00:30:15 --> 00:30:18
formula for area here.
541
00:30:18 --> 00:30:22
Because this is a really
beautiful connection.
542
00:30:22 --> 00:30:26
And I want to make that
connection in connection
543
00:30:26 --> 00:30:29
also with this example.
544
00:30:29 --> 00:30:33
The hyperbolas, as you
probably know, are the
545
00:30:33 --> 00:30:37
trajectories of comets.
546
00:30:37 --> 00:30:42
And ellipses, which is what you
would get if maybe you put 1/2
547
00:30:42 --> 00:30:44
here instead of a 2, would
be the trajectories of
548
00:30:44 --> 00:30:47
planets or asteroids.
549
00:30:47 --> 00:30:51
But there's actually something
much more important,
550
00:30:51 --> 00:30:53
physically that goes on.
551
00:30:53 --> 00:30:56
That's special about this
particular representation
552
00:30:56 --> 00:30:58
of the hyperbola.
553
00:30:58 --> 00:31:01
And what happens when you
get the ellipses as well.
554
00:31:01 --> 00:31:08
Which is that in this
case, r = 0 is the
555
00:31:08 --> 00:31:17
focus of the hyperbola.
556
00:31:17 --> 00:31:22
And what that means is that
it's actually the place
557
00:31:22 --> 00:31:28
where the sun is.
558
00:31:28 --> 00:31:31
So this is the right
representation, if you want
559
00:31:31 --> 00:31:36
the center of gravity in
the center of your picture.
560
00:31:36 --> 00:31:38
And pretty much any other.
561
00:31:38 --> 00:31:40
I mean, you can't tell that
at all from the algebraic
562
00:31:40 --> 00:31:43
equations here.
563
00:31:43 --> 00:31:46
So this hyperbola is going to
be the trajectory of some
564
00:31:46 --> 00:31:51
comet going by here.
565
00:31:51 --> 00:31:58
And this formula here is
actually a rather central
566
00:31:58 --> 00:32:05
formula in astronomy.
567
00:32:05 --> 00:32:17
Namely, there's something
called Kepler's Law.
568
00:32:17 --> 00:32:23
Which says that the rate of
change of area which is
569
00:32:23 --> 00:32:28
swept out is constant.
570
00:32:28 --> 00:32:30
The rate of change of area
relative to the center of
571
00:32:30 --> 00:32:33
mass, relative to the sun.
572
00:32:33 --> 00:32:37
So in equal areas, this
is amount of area.
573
00:32:37 --> 00:32:43
So this tells you now that when
a comet goes around the sun
574
00:32:43 --> 00:32:46
like this, its speed varies.
575
00:32:46 --> 00:32:49
And it's speed varies according
to a very specific rule.
576
00:32:49 --> 00:32:52
Namely, this one here.
577
00:32:52 --> 00:32:54
And this rule was
observed by Kepler.
578
00:32:54 --> 00:32:58
But if you have this
connection here, we also
579
00:32:58 --> 00:32:59
have something else.
580
00:32:59 --> 00:33:08
We also know that dA / dt
= 1/2 r ^2 d theta / dt.
581
00:33:08 --> 00:33:13
So that's this formula here,
formally dividing by t.
582
00:33:13 --> 00:33:16
That's the rate of change with
respect to this time parameter,
583
00:33:16 --> 00:33:18
which is the honest
to goodness time.
584
00:33:18 --> 00:33:20
Real physical time.
585
00:33:20 --> 00:33:34
And that means, this
quantity here is constant.
586
00:33:34 --> 00:33:40
And this is one of the key
insights that physicists had,
587
00:33:40 --> 00:33:43
long after Kepler made his
physical observations, they
588
00:33:43 --> 00:33:48
realized that he had managed to
get the best physics experiment
589
00:33:48 --> 00:33:50
of all, because it's a
frictionless setup.
590
00:33:50 --> 00:33:53
Outer space, there's no air.
591
00:33:53 --> 00:33:54
Nothing is going on.
592
00:33:54 --> 00:33:59
This is what's known
nowadays as conservation
593
00:33:59 --> 00:34:10
of angular momentum.
594
00:34:10 --> 00:34:13
This is the expression
for angular momentum.
595
00:34:13 --> 00:34:18
And what Kepler was observing,
it turns out, is what we see
596
00:34:18 --> 00:34:19
all the time in real life.
597
00:34:19 --> 00:34:21
Which is when you start
something spinning around
598
00:34:21 --> 00:34:23
it continues to spin at
roughly the same rate.
599
00:34:23 --> 00:34:27
Or, if you're an ice skater and
you get yourself scrunched
600
00:34:27 --> 00:34:30
together a little bit more,
you can spin faster.
601
00:34:30 --> 00:34:32
And there's an exact
quantitative rule
602
00:34:32 --> 00:34:33
that does that.
603
00:34:33 --> 00:34:38
And it's exactly this
polar formula here.
604
00:34:38 --> 00:34:39
So that's a neat thing.
605
00:34:39 --> 00:34:41
And we will do a little
exercise on this rate of
606
00:34:41 --> 00:34:45
change after the exam.
607
00:34:45 --> 00:34:52
So that's it for generalities
and a little pep talk on what's
608
00:34:52 --> 00:34:57
coming up to you when you
learn a little more physics.
609
00:34:57 --> 00:35:04
Right now we need to
talk about the exam.
610
00:35:04 --> 00:35:14
So first of all, let me tell
you what the topics are.
611
00:35:14 --> 00:35:17
They're the same as
last year's test.
612
00:35:17 --> 00:35:19
Which you can take a look at.
613
00:35:19 --> 00:35:24
And let's see.
614
00:35:24 --> 00:35:25
So what did we do?
615
00:35:25 --> 00:35:30
One of the main topics of
this unit were techniques
616
00:35:30 --> 00:35:35
of integration.
617
00:35:35 --> 00:35:40
And there are three,
which we will test.
618
00:35:40 --> 00:35:46
One is trig substitution.
619
00:35:46 --> 00:35:53
One is integration by parts.
620
00:35:53 --> 00:36:02
And one is partial fractions.
621
00:36:02 --> 00:36:06
So that's more than half
of the exam, right there.
622
00:36:06 --> 00:36:16
The other half of the exam
is parametric curves.
623
00:36:16 --> 00:36:18
Arc length.
624
00:36:18 --> 00:36:20
These are all interrelated.
625
00:36:20 --> 00:36:33
And area of surfaces
of revolution.
626
00:36:33 --> 00:36:35
Those are the only kind
that we can handle.
627
00:36:35 --> 00:36:38
Just as we did with volume
of surfaces of revolution.
628
00:36:38 --> 00:36:47
And then there's a final topic,
which is polar coordinates.
629
00:36:47 --> 00:36:57
And area in polar
coordinates, including area.
630
00:36:57 --> 00:36:57
That's it.
631
00:36:57 --> 00:37:00
That's what's on the test,
there are six problems.
632
00:37:00 --> 00:37:02
They're very similar.
633
00:37:02 --> 00:37:04
Well, they're not
actually that similar.
634
00:37:04 --> 00:37:07
But they're somewhat
similar to last year's.
635
00:37:07 --> 00:37:13
I'd say the test is similar.
636
00:37:13 --> 00:37:15
Maybe a tiny bit
more difficult.
637
00:37:15 --> 00:37:16
We'll see.
638
00:37:16 --> 00:37:18
We'll see.
639
00:37:18 --> 00:37:18
Yeah.
640
00:37:18 --> 00:37:23
STUDENT: [INAUDIBLE]
641
00:37:23 --> 00:37:26
PROFESSOR: The question was,
we didn't do arc length in
642
00:37:26 --> 00:37:28
polar coordinates, did we?
643
00:37:28 --> 00:37:31
And the answer is
no, we did not.
644
00:37:31 --> 00:37:32
We did not do arc length
in polar coordinates.
645
00:37:32 --> 00:37:36
When I give you an exercise,
I'm going to ask you about, if
646
00:37:36 --> 00:37:38
you know the speed of a comet
here, what's the speed
647
00:37:38 --> 00:37:39
of the comet there.
648
00:37:39 --> 00:37:41
And we'll have to know
about arc length for that.
649
00:37:41 --> 00:37:48
But we're not doing
it on this exam.
650
00:37:48 --> 00:37:49
Other questions.
651
00:37:49 --> 00:37:59
STUDENT: [INAUDIBLE]
652
00:37:59 --> 00:38:05
PROFESSOR: The question is,
will I expect you to know r
653
00:38:05 --> 00:38:10
equals, so let's see if I can
formulate this question.
654
00:38:10 --> 00:38:13
It's related to this
four-leaf rose here.
655
00:38:13 --> 00:38:16
So the question is, suppose
I gave you something
656
00:38:16 --> 00:38:21
that looked like this.
657
00:38:21 --> 00:38:24
Would I expect you to be
able to know what it is.
658
00:38:24 --> 00:38:28
I think the answer, the fair
answer to give you, is if it's
659
00:38:28 --> 00:38:33
this complicated, I only
have two possibilities.
660
00:38:33 --> 00:38:37
I can give you a long
time to sketch this out.
661
00:38:37 --> 00:38:38
And think about what it does.
662
00:38:38 --> 00:38:40
Or I can tell you that
it happens to be a
663
00:38:40 --> 00:38:42
three-leaf rose.
664
00:38:42 --> 00:38:47
And then you have some clue
as to what it's doing.
665
00:38:47 --> 00:38:49
It doesn't have six.
666
00:38:49 --> 00:38:53
Because of some weird thing,
having to do with repetitions.
667
00:38:53 --> 00:38:57
But the odds and evens
work differently.
668
00:38:57 --> 00:39:03
So, in fact I would have to
tell you what the picture
669
00:39:03 --> 00:39:07
looks like, if it's going
to be this complicated.
670
00:39:07 --> 00:39:11
Similarly, so this is an
important point to make, when
671
00:39:11 --> 00:39:13
we come to techniques of
integration, any integral that
672
00:39:13 --> 00:39:18
you have, I'm not going to tell
you which of these three
673
00:39:18 --> 00:39:20
techniques to use on the ones
which are straightforward
674
00:39:20 --> 00:39:20
integrals.
675
00:39:20 --> 00:39:22
But if it's an integral that I
think you're going to get stuck
676
00:39:22 --> 00:39:25
on, either I'm going to
give you a hint, tell
677
00:39:25 --> 00:39:26
you how to do it.
678
00:39:26 --> 00:39:29
Or I'm going to tell
you, don't do it.
679
00:39:29 --> 00:39:31
If I tell you don't do
it, don't try to do it.
680
00:39:31 --> 00:39:33
It may be impossible.
681
00:39:33 --> 00:39:35
And even if it's possible,
it's going to be very long.
682
00:39:35 --> 00:39:38
Like an hour.
683
00:39:38 --> 00:39:42
So don't do it unless
I tell you to.
684
00:39:42 --> 00:39:46
On the other hand, all of these
setups in this second half
685
00:39:46 --> 00:39:50
of this unit, they involve
somehow setting something up.
686
00:39:50 --> 00:39:55
And they're basically
three issues.
687
00:39:55 --> 00:39:57
One is what the integrand is.
688
00:39:57 --> 00:40:02
One is what the lower limit
is, what is the upper limit.
689
00:40:02 --> 00:40:05
They're just three
things, three inputs, to
690
00:40:05 --> 00:40:06
setting up an integral.
691
00:40:06 --> 00:40:10
All integrals, this is going to
be the setup for all of them.
692
00:40:10 --> 00:40:13
And then the second
step is evaluating.
693
00:40:13 --> 00:40:17
Which really is what we did
in the first half here.
694
00:40:17 --> 00:40:20
And, unfortunately, we don't
have infinitely many techniques
695
00:40:20 --> 00:40:21
and indeed there's some
integrals that can't be
696
00:40:21 --> 00:40:23
evaluated and some
that are too long.
697
00:40:23 --> 00:40:25
So we'll just try
to avoid those.
698
00:40:25 --> 00:40:32
I'm not trying to give you ones
which are hopelessly long.
699
00:40:32 --> 00:40:33
Alright, other questions.
700
00:40:33 --> 00:40:34
Yes.
701
00:40:34 --> 00:40:47
STUDENT: [INAUDIBLE]
702
00:40:47 --> 00:40:49
PROFESSOR: The question
is, will the percentages
703
00:40:49 --> 00:40:50
be the same.
704
00:40:50 --> 00:40:54
And the answer is, no.
705
00:40:54 --> 00:40:55
I'll tell you exactly.
706
00:40:55 --> 00:40:57
This is 55 points.
707
00:40:57 --> 00:40:59
Unless I change
the point values.
708
00:40:59 --> 00:41:07
This is 55, and this is 45.
709
00:41:07 --> 00:41:11
That's what it came out to be.
710
00:41:11 --> 00:41:14
You are going to want to know
about all of the things that
711
00:41:14 --> 00:41:14
I've written down here.
712
00:41:14 --> 00:41:17
You're definitely going to
want to know, for example,
713
00:41:17 --> 00:41:18
surfaces of revolution.
714
00:41:18 --> 00:41:20
How to set those up.
715
00:41:20 --> 00:41:24
Yes. there was another
question I saw.
716
00:41:24 --> 00:41:24
Yes.
717
00:41:24 --> 00:41:50
STUDENT: [INAUDIBLE]
718
00:41:50 --> 00:41:55
PROFESSOR: So if you have a
partial fraction with something
719
00:41:55 --> 00:42:11
like (x + 2)^2 and maybe an x
and maybe an x + 1, and you're
720
00:42:11 --> 00:42:16
interested in what happens
with this denominator here?
721
00:42:16 --> 00:42:24
So what's going to happen
is, you're going to
722
00:42:24 --> 00:42:32
need a coefficient for
each degree of this.
723
00:42:32 --> 00:42:40
So altogether, the setup
is going to be this.
724
00:42:40 --> 00:42:43
Plus one for x.
725
00:42:43 --> 00:42:46
And one for x + 1.
726
00:42:46 --> 00:42:50
This is the setup.
727
00:42:50 --> 00:42:51
So you need --
728
00:42:51 --> 00:42:58
STUDENT: [INAUDIBLE]
729
00:42:58 --> 00:43:02
PROFESSOR: So if I change this
to being a 3 here, then I
730
00:43:02 --> 00:43:10
need, I guess I'll have
to call it E, (x + 2) ^3.
731
00:43:10 --> 00:43:11
I need that.
732
00:43:11 --> 00:43:13
Now, it gets harder and harder.
733
00:43:13 --> 00:43:15
The more repeated roots there
are, the more repeated factors
734
00:43:15 --> 00:43:17
there are, the harder it is.
735
00:43:17 --> 00:43:22
Because the ones you can pick
off by the cover-up method are,
736
00:43:22 --> 00:43:24
is just the top one here.
737
00:43:24 --> 00:43:25
And these two.
738
00:43:25 --> 00:43:28
So C, D, and E you can get.
739
00:43:28 --> 00:43:31
But B and A you're going to
have to do by either plugging
740
00:43:31 --> 00:43:33
in or some other, more
elaborate, algebra.
741
00:43:33 --> 00:43:36
So the more of these lower
terms there are, the
742
00:43:36 --> 00:43:37
worse off you are.
743
00:43:37 --> 00:43:48
STUDENT: [INAUDIBLE]
744
00:43:48 --> 00:43:56
PROFESSOR: The question
is, does this x ^3 +
745
00:43:56 --> 00:43:59
21 affect this setup.
746
00:43:59 --> 00:44:03
And the answer is almost no.
747
00:44:03 --> 00:44:04
That is, not at all.
748
00:44:04 --> 00:44:06
It's the same setup exactly.
749
00:44:06 --> 00:44:08
But, there's one thing.
750
00:44:08 --> 00:44:12
If the degree gets too big,
then you've got to use long
751
00:44:12 --> 00:44:18
division first to
knock it down.
752
00:44:18 --> 00:44:20
I'll give you an example
of this type of practice.
753
00:44:20 --> 00:44:22
Unless there are more question.
754
00:44:22 --> 00:44:22
Yes.
755
00:44:22 --> 00:44:23
STUDENT: [INAUDIBLE]
756
00:44:23 --> 00:44:26
PROFESSOR: Are you going
to have to know how to
757
00:44:26 --> 00:44:29
do reduction formulas?
758
00:44:29 --> 00:44:33
Anything that's a little out of
the ordinary like a reduction
759
00:44:33 --> 00:44:35
formula, I will have
to coach you to do.
760
00:44:35 --> 00:44:38
So, in other words, what you'll
have to be able to do in that
761
00:44:38 --> 00:44:40
situation is follow directions.
762
00:44:40 --> 00:44:42
If I tell you OK, you're
faced with this, then do
763
00:44:42 --> 00:44:44
an integration by parts.
764
00:44:44 --> 00:44:48
And do that, then get
the reduction formula.
765
00:44:48 --> 00:44:49
STUDENT: [INAUDIBLE]
766
00:44:49 --> 00:44:50
PROFESSOR: Yeah.
767
00:44:50 --> 00:44:56
OK, so the question had do with
the partial fractions method.
768
00:44:56 --> 00:45:03
And what happens if
you have a quadratic.
769
00:45:03 --> 00:45:12
So, for instance, if it
were this, this one's
770
00:45:12 --> 00:45:13
too disgusting.
771
00:45:13 --> 00:45:17
I'm going to just do
it with two of them.
772
00:45:17 --> 00:45:20
So the parts with x +
x + 1 are the same.
773
00:45:20 --> 00:45:23
But now you have
linear factors here.
774
00:45:23 --> 00:45:27
Ax + B / x^2 + 2.
775
00:45:27 --> 00:45:33
And A, maybe I'll call them
1, and A2 x + B2 / (x^2 +
776
00:45:33 --> 00:45:43
2) ^2 + C / x + D / x + 1.
777
00:45:43 --> 00:45:47
This is the way it works.
778
00:45:47 --> 00:45:52
OK, I'm going to give you one
more quick example of an
779
00:45:52 --> 00:45:59
integration technique
just to liven things up.
780
00:45:59 --> 00:46:02
Let's see.
781
00:46:02 --> 00:46:06
So here's a somewhat
tricky example.
782
00:46:06 --> 00:46:08
This is just a little
trickier than I would
783
00:46:08 --> 00:46:09
give you on a test.
784
00:46:09 --> 00:46:11
But it's the same principle,
and I may do this
785
00:46:11 --> 00:46:14
on a final exam.
786
00:46:14 --> 00:46:20
So suppose you're faced
with this integral.
787
00:46:20 --> 00:46:24
What are you going to do?
788
00:46:24 --> 00:46:26
Integration by parts, great.
789
00:46:26 --> 00:46:29
That's right, that's because
this guy is begging to be
790
00:46:29 --> 00:46:33
differentiated, to
be made simpler.
791
00:46:33 --> 00:46:37
So that means that I want
this one to be u, and I
792
00:46:37 --> 00:46:40
want this one to be v '.
793
00:46:40 --> 00:46:42
And I want to use
integration by parts.
794
00:46:42 --> 00:46:51
And then u ' = 1 / 1 + x
^2, and v = x ^2 / 2.
795
00:46:51 --> 00:46:58
So the answer is now, x^2
/ 2 tan inverse x - the
796
00:46:58 --> 00:47:00
integral of this guy.
797
00:47:00 --> 00:47:02
Which is going to be x ^2 / 2.
798
00:47:02 --> 00:47:09
And then I have 1
/ 1 + x ^2 dx.
799
00:47:09 --> 00:47:13
Now, you are not
done at this point.
800
00:47:13 --> 00:47:16
You're still in
slightly hot water.
801
00:47:16 --> 00:47:19
You're in tepid water, anyway.
802
00:47:19 --> 00:47:21
So what is it that
you have to do here?
803
00:47:21 --> 00:47:24
You're faced with this
integral, which I'll
804
00:47:24 --> 00:47:30
put on the next board.
805
00:47:30 --> 00:47:33
It's a lot simpler than the
other one, but as I say you're
806
00:47:33 --> 00:47:35
not quite out of the woods.
807
00:47:35 --> 00:47:39
You're faced with the
integral of 1/2, - 1/2
808
00:47:39 --> 00:47:42
x ^2 / 1 + x ^2 dx.
809
00:47:42 --> 00:47:47
STUDENT: [INAUDIBLE]
810
00:47:47 --> 00:47:50
PROFESSOR: Trig substitution
actually, interestingly,
811
00:47:50 --> 00:47:53
will work.
812
00:47:53 --> 00:47:54
But that wasn't what
I wanted you to do.
813
00:47:54 --> 00:47:56
I wanted you to,
yeah, go ahead.
814
00:47:56 --> 00:47:57
STUDENT: [INAUDIBLE]
815
00:47:57 --> 00:47:59
PROFESSOR: Add and subtract
1 to the numerator.
816
00:47:59 --> 00:48:02
So now, that's the
correct answer.
817
00:48:02 --> 00:48:04
This is the case where
the numerator and the
818
00:48:04 --> 00:48:06
denominator are tied.
819
00:48:06 --> 00:48:08
And so you have to
use long division.
820
00:48:08 --> 00:48:12
But a shortcut is just to
observe that the result of
821
00:48:12 --> 00:48:20
long division is the same
thing as doing this.
822
00:48:20 --> 00:48:26
And then noticing that
this is 1 - 1 / 1 + x ^2.
823
00:48:26 --> 00:48:30
So this is the same as long
division, in this case.
824
00:48:30 --> 00:48:34
Because when you divide in, it
goes in with a quotient of 1.
825
00:48:34 --> 00:48:39
And so this guy turns out
to be - 1/2 the integral
826
00:48:39 --> 00:48:47
of 1 - 1 / 1 + x ^2 dx.
827
00:48:47 --> 00:48:55
Which is 1/2 x - 1/2
tan inverse x + c.
828
00:48:55 --> 00:49:01
So this is one extra step
that you may be faced with
829
00:49:01 --> 00:49:02
someday in your life.
830
00:49:02 --> 00:49:05
And just keep that in mind.
831
00:49:05 --> 00:49:06