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CHRISTINE BREINER: Welcome
back to recitation.

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In this video I'd like
us to do two things.

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The first thing
we're going to do

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is we're going to
graph the curve

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r equals 1 plus
cosine theta over 2,

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for theta between 0 and
4 pi, and we're going

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to graph it in the xy-plane.

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And then after we've
done that, we're

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going to take a look at some
components of that curve

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and we're going to calculate
the area of some components that

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close up.

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So what I'd like you to do
first is get a good picture

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of this curve, in the xy-plane.

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I'll give you a little
while to do that.

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So why don't you
pause the video,

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get a good picture
of that curve,

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then come back when you're
ready and I'll show you

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how I graph it, and then we'll
get into these area problems.

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OK.

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Welcome back.

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So the goal, again, was to graph
a certain curve described by r

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and theta, but in the xy-plane.

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For theta between
0 and pi over 4.

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And when I do these
problems, we want

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to make sure that we understand
how r depends on theta.

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That's sort of the main
goal to graph this curve.

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So what I do,
actually, is I look

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at this not in the
xy-plane, which

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was what I said to
do in the problem,

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but I first look at it in what
we see as the r, theta plane.

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And what we mean
by that is we're

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going to graph this just like
we would if this variable was x

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and this variable was y.

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So we move out of what we know
about how r relates to x and y,

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and how theta
relates to x and y,

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and we just look at
how r relates to theta.

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So let me draw that
first and we'll

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see if we can sort of
understand what I mean by that,

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and then we'll put that
picture, use that curve

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to put that into the xy-plane.

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OK.

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So, the first thing
we do is I'm going

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to let this be the theta-axis
and this be the r-axis.

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And let's look at--

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I want to write down over here
what the equation is so I don't

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have to keep turning around.

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So again, this is
not, I don't want

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to think about this
as the xy-plane.

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Because in the
xy-plane, theta has

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certain values at
each point that

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are fixed based on the angle.

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But now I'm letting theta
vary in this direction,

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and r is varying
in this direction.

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And my theta, I said,
was between 0 and pi

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over 4-- I'm sorry,
not pi over 4.

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4*pi.

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0 and 4*pi.

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pi over 4 would be a very
small component of this

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that I'm interested in.

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And let's think
about what happens,

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what kind of transformations
have been done

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to the normal cosine function.

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So if I took the
normal cosine function

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and I take it-- instead
of cosine theta,

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I look it cosine theta
over 2, what that's doing

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is that's stretching
it horizontally out.

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So think about the period of
the cosine function usually is

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2*pi.

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But notice what happens when
I put in 2*pi for theta,

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I'm getting cosine of pi.

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If I want to get cosine of 2*pi,
I have to let theta go to 4*pi,

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which is why I'm letting
theta be between 0 and 4*pi.

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So dividing the input value
by 2 doubles your period.

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So the period is now 4*pi.

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So to get all the way through,
I'm going to have to go up

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to 4*pi.

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So let me just make a 2*pi here,
and this is about a 4*pi here.

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So here's around 3 pi,
and here's around pi.

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So that now, instead of
the usual cosine function,

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it's going to take twice as
long to get all the way through.

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That's one thing we know.

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What else have we done to
the usual cosine function?

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We've moved it up by 1.

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And so instead of starting
out when your input is 0,

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starting out at height 1,
when you're input is 0,

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you start out at
height 1 plus 1.

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You start out at height 2.

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So in fact, this function,
let me point out,

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it's going to start at
2, which means it also

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is going to end over here.

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Because it's periodic,
it's going to end at 2.

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And let's think about
what else we know.

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We know that the usual
cosine function goes down

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to negative 1.

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But I've added 1 to it, so
now it only goes down to 0.

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OK?

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Hopefully that makes sense.

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Maybe I should even-- hmm.

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I don't want to draw
the actual cosine

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function again right on here.

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But let me draw the regular
cosine function here.

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So we have it,
the regular cosine

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function-- because I
keep talking about it--

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does something like this.

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It goes at 1 here, and
it's at 1 again here.

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And so it's at minus 1 at pi.

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And so, very roughly, it
looks something like this.

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Right?

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So I keep referencing
the cosine function,

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so this is the part
I'm referencing.

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So we have to stretch it by
2, and then shift it up by 1.

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And so we see what
was at pi, negative 1,

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I'm now going to be at 2*pi, 0.

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And then where do
these points go?

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This was pi over 2.

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The x-value is
going to be doubled.

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I'm going to be at pi, and the
y-value is going to go up by 1.

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So I'm at (pi, 1) and (3*pi, 1).

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And so the curve will
look something like this.

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I'm not an expert here,
but hopefully that

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looks something like
the cosine function.

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But with a stretch and a shift.

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So that is the curve r equals
1 plus cosine theta over 2

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in what we consider
the r, theta plane.

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So theta is varying
in this direction

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and r is varying
in this direction.

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But how do I transfer
that to the xy-plane?

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That's the real
point that I want

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to make about this problem.

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So let's look at what's
happening in the xy-plane.

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So this will be x
and this will be y.

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Let's pick some points and try
to figure out what's happening.

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So where is this point?

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This is the point
zero comma two.

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When theta is 0, r equals 2.

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Where is theta equal
to 0 in this picture?

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It's in the x-direction.

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Right?

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That's theta equals 0,
and also 2*pi and 4*pi.

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Any multiple of 2*pi, theta
is pointing in this direction.

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And r equals 2 there.

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So in a strange twist,
this is the point (2, 0)

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on the x, y plane.

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But that's no going to,
that's just a coincidence, OK?

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Don't think, oh, I'm just going
to flip the values everywhere.

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That's not going to happen.

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OK?

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Actually, and also,
before I make a mistake,

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I'm going to make this
a little bit bigger.

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I want it to be bigger.

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So I don't want this to be 2.

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I want this to be 1.

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I'm going to make this (2, 0).

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I want to have a
little bigger picture.

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OK.

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So it's going to be (2, 0) at
theta equals 0 and r equal 2.

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Is it ever hit this point again?

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Well, it is.

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And it's going to hit that point
again because it's periodic

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and I've gone out to 4*pi.

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If I rotate all the way
around, I'm at 2*pi for theta.

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If I rotate all the way around
again I'm at 4*pi for theta,

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and my radius there is 2, also.

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So this (2, 0) happens again
when I have this point.

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So it's going to close up.

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OK.

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And then what else happens?

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Well, as theta is
rotating, let's

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take theta between 0 and pi.

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theta rotates from 0
to pi going like this.

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Notice what's happening
to the r-value.

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The r-value is going
from 2 down to 1.

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Now, I'm not going
to be totally exact,

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but-- here's minus 1,
OK, in the xy-plane.

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I'm not going to
be totally exactly,

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but this curve is
going to look something

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like-- there's 2-- it's going
to look something like-- oops,

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I overshot, we'll make
that negative 1-- something

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like this.

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And what's the point?

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The point is that I start
at radius 2, and by the time

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I get to theta equals
pi I've gone down.

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And so my radius is 1.

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This has radius 1 and angle pi.

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So that represents
this part of the curve.

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That's this part of the curve.

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Now, what's nice
about this drawing

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is that we know this
part of the curve

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and this part of the curve
should look exactly the same.

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So once I've drawn half of this,
I'm going to know everything.

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Once I've gone from 0 to 2*pi, I
can just reflect it, basically.

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We'll see what I mean
by reflect in this case,

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but the same radii are
happening again and some sort

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of symmetric fashion.

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OK.

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So we've got 0 to pi.

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Now what happens
between pi and 2*pi?

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Notice pi to 2*pi in the theta
direction on the xy-plane is I

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start in this direction-- I
don't know if you can see that,

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let me come over to this side--
I start in this direction

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for pi, and I'm
going to rotate down.

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This is 3*pi over
2, and this is 2*pi.

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Those are my angles.

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So what are my radii?

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Well, I start at radius 1
and I'm going to radius 0.

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And so what happens is I'm
coming through this negative 1

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and I'm coming around, and
then by the time I get--

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it's going to be
something like this--

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by the time I get to
2*pi, my radius is 0.

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Now just to make sure this
curve makes sense to you,

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you could pick a place, an
angle, maybe like right here.

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I don't know if
that's easy to see,

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but maybe that
angle right there.

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That angle is between
3*pi over 2 and 2*pi.

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Is there positive radius there?

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00:10:01,280 --> 00:10:02,900
Yeah, there is
positive radius there.

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So in fact, this curve does
come into this fourth quadrant

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here and then curve back in.

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OK?

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It does curve back in.

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All right.

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So then what happens
between 2*pi and 3*pi?

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OK.

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Hopefully this picture
is clear so far.

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I'm going to come back
to the other side.

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OK.

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What happens between
2*pi and 3*pi?

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2*pi, we're here.

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And 3*pi, we're back
over here again.

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And notice that the radius is
going to be doing the same kind

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of growth that it did between
2*pi and 3*pi as it did decay

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from pi to 2*pi.

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OK?

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Because the radii now,
there's a symmetry

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with how the radii behave.

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So from 2*pi to 3*pi, I start
off with radius 0 and I have

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a small radius.

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And then once I get
to 3*pi over here,

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I'm going to have
radius 1 again.

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I'm going to be at
radius 1, which is going

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to correspond to this point.

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So I'm going to have exactly
the same picture, which

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is dangerous because I
probably would do it wrong.

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But I'll try and draw it this
way and then talk about it.

245
00:11:06,840 --> 00:11:09,480
Hopefully that looks
about the same.

246
00:11:09,480 --> 00:11:13,570
So this curve coming through
here was from pi to 2*pi.

247
00:11:13,570 --> 00:11:17,750
This curve coming through
here was from to 2*pi to 3*pi.

248
00:11:17,750 --> 00:11:20,470
And then to finish it off, 3*pi
to 4*pi is going to look like

249
00:11:20,470 --> 00:11:22,310
this curve here.

250
00:11:22,310 --> 00:11:26,150
So I come through-- ooh, this
is where it starts to get really

251
00:11:26,150 --> 00:11:29,980
dangerous, but let's say
that's pretty close--

252
00:11:29,980 --> 00:11:33,160
so there's my 3*pi to 4*pi.

253
00:11:33,160 --> 00:11:35,660
It's again, the
same growth the way

254
00:11:35,660 --> 00:11:38,260
it was decaying
between 0 and pi.

255
00:11:38,260 --> 00:11:44,360
So this is your picture in the
xy-plane of the curve r equals

256
00:11:44,360 --> 00:11:47,150
1 plus cosine theta over 2.

257
00:11:47,150 --> 00:11:50,900
Now, we haven't calculated
any area problems, yet.

258
00:11:50,900 --> 00:11:55,470
So what I'd like us to do is
I'm going to shade two regions,

259
00:11:55,470 --> 00:12:00,430
and I want us to just write
down the integral form

260
00:12:00,430 --> 00:12:04,070
to find the area for
each of these regions.

261
00:12:04,070 --> 00:12:09,600
So the first region of
interest is the pink region.

262
00:12:09,600 --> 00:12:12,750
I'm going to ask us to find
the area of the pink region,

263
00:12:12,750 --> 00:12:16,420
and then I'm going to ask us
to find the area of everything

264
00:12:16,420 --> 00:12:17,435
else, the blue region.

265
00:12:20,450 --> 00:12:24,480
So let's think about
how to do that.

266
00:12:24,480 --> 00:12:26,980
And I think I'm going to have
to come over to the other side

267
00:12:26,980 --> 00:12:30,440
to write down the integrals, but
I'll be coming back and forth.

268
00:12:30,440 --> 00:12:32,660
So just to remind
you, what you saw

269
00:12:32,660 --> 00:12:40,550
in lecture was that dA is equal
to r squared over 2 d theta.

270
00:12:40,550 --> 00:12:42,740
That's what dA is.

271
00:12:42,740 --> 00:12:45,960
And so this is going to
be an integral in theta,

272
00:12:45,960 --> 00:12:48,650
and we know what r is
as a function of theta.

273
00:12:48,650 --> 00:12:50,150
And so if I want
to find-- actually,

274
00:12:50,150 --> 00:12:54,330
I should even use my
colors appropriately.

275
00:12:54,330 --> 00:12:59,710
I should say, the
pink area is going

276
00:12:59,710 --> 00:13:01,860
to be equal to an integral.

277
00:13:01,860 --> 00:13:03,780
And I'm going to
write the r squared.

278
00:13:03,780 --> 00:13:05,010
I know what r squared is.

279
00:13:05,010 --> 00:13:10,200
It's 1 plus cosine
theta over 2 squared,

280
00:13:10,200 --> 00:13:12,560
and then an over 2 d theta.

281
00:13:12,560 --> 00:13:14,810
And then what is
important about this?

282
00:13:14,810 --> 00:13:15,590
It's our bounds.

283
00:13:15,590 --> 00:13:16,090
Right?

284
00:13:16,090 --> 00:13:17,800
Our bounds are what's important.

285
00:13:17,800 --> 00:13:21,760
And so let's go back and
let's look at our picture.

286
00:13:21,760 --> 00:13:25,680
What are the bounds on
theta for the pink region?

287
00:13:25,680 --> 00:13:28,540
So where does that pink
region start and stop?

288
00:13:28,540 --> 00:13:32,530
And maybe we even need to
look at this top graph, also.

289
00:13:32,530 --> 00:13:36,280
So if we think about
it, we went from 0 to pi

290
00:13:36,280 --> 00:13:38,066
to get this outer curve.

291
00:13:38,066 --> 00:13:39,440
So how do we get
the inner curve?

292
00:13:39,440 --> 00:13:44,070
The inner curve started at
theta equals pi, went to here,

293
00:13:44,070 --> 00:13:47,900
went to here-- that
was theta equals 3*pi.

294
00:13:47,900 --> 00:13:52,160
So it went all the
way from pi to 3*pi.

295
00:13:52,160 --> 00:13:54,780
Now, if you're paying
good attention,

296
00:13:54,780 --> 00:13:58,300
you can say, well, Christine,
we know that this region is

297
00:13:58,300 --> 00:13:59,550
totally symmetric.

298
00:13:59,550 --> 00:14:02,730
So why don't I just take
the area from pi to 2*pi

299
00:14:02,730 --> 00:14:04,470
and multiply it by 2?

300
00:14:04,470 --> 00:14:05,340
And you can.

301
00:14:05,340 --> 00:14:06,910
You can do it either way.

302
00:14:06,910 --> 00:14:09,900
So you can either take the
integral from all the way from

303
00:14:09,900 --> 00:14:13,880
pi to 3*pi, which corresponds
to starting at this angle,

304
00:14:13,880 --> 00:14:16,324
going all the way around,
and coming back to there,

305
00:14:16,324 --> 00:14:18,240
which takes you all the
way around this curve.

306
00:14:18,240 --> 00:14:21,780
Or you can go from pi to
2*pi and multiply that by 2.

307
00:14:21,780 --> 00:14:25,470
So let me come back and
let me write that down.

308
00:14:25,470 --> 00:14:32,430
It's either pi to 3*pi, this,
or you just write it as integral

309
00:14:32,430 --> 00:14:34,310
from pi to 2*pi.

310
00:14:34,310 --> 00:14:38,430
And if I multiply this
by 2, the 2 drops out.

311
00:14:38,430 --> 00:14:40,477
The 2 in the
denominator drops out.

312
00:14:44,407 --> 00:14:46,240
I'm not going to solve
this problem for you,

313
00:14:46,240 --> 00:14:48,781
but I do want to point out the
kinds of terms you would have.

314
00:14:48,781 --> 00:14:51,060
You would have a constant
term when you square this.

315
00:14:51,060 --> 00:14:54,150
You would have a term that
was 2 cosine theta over 2,

316
00:14:54,150 --> 00:14:56,980
which is easy to integrate
by a u-substitution.

317
00:14:56,980 --> 00:14:59,450
And then you would have
a cosine squared theta

318
00:14:59,450 --> 00:15:02,460
over 2, which you'd want
to use the double angle

319
00:15:02,460 --> 00:15:03,940
formula or the
half angle formula

320
00:15:03,940 --> 00:15:07,060
that you've seen used to
manipulate these integrals that

321
00:15:07,060 --> 00:15:09,680
involve just a cosine
squared or a sine squared.

322
00:15:09,680 --> 00:15:11,600
So that would be your strategy.

323
00:15:11,600 --> 00:15:13,175
OK, now let's look
at the blue area.

324
00:15:13,175 --> 00:15:13,675
OK.

325
00:15:19,090 --> 00:15:20,840
So to find the
blue area, again, I

326
00:15:20,840 --> 00:15:24,307
know all that matters
is really the bounds.

327
00:15:24,307 --> 00:15:26,515
We're going to see we have
to do a little extra work.

328
00:15:29,600 --> 00:15:31,000
But this is our first setup.

329
00:15:31,000 --> 00:15:32,861
And now let's go
look at the bounds.

330
00:15:32,861 --> 00:15:33,360
OK.

331
00:15:33,360 --> 00:15:35,880
So we go back to the curve.

332
00:15:35,880 --> 00:15:37,020
All right.

333
00:15:37,020 --> 00:15:38,000
What is the blue area?

334
00:15:38,000 --> 00:15:41,730
Well, the blue area,
that's a little harder.

335
00:15:41,730 --> 00:15:43,680
So let's see what happens.

336
00:15:43,680 --> 00:15:49,490
If I were to take theta from
0 to pi, what would happen?

337
00:15:49,490 --> 00:15:51,460
I would not only pick
up the blue area,

338
00:15:51,460 --> 00:15:54,100
but I'd pick up this
pink stuff inside.

339
00:15:54,100 --> 00:15:55,640
But I don't want the pink stuff.

340
00:15:55,640 --> 00:15:57,290
I just want the blue stuff.

341
00:15:57,290 --> 00:16:00,020
So what am I going to have to
do to find the blue area just,

342
00:16:00,020 --> 00:16:01,980
say, from 0 up to pi?

343
00:16:01,980 --> 00:16:04,641
I'm going to have to find
the area from 0 to pi,

344
00:16:04,641 --> 00:16:06,390
and then I'm going to
have to subtract off

345
00:16:06,390 --> 00:16:07,722
the area of this component.

346
00:16:07,722 --> 00:16:09,184
OK?

347
00:16:09,184 --> 00:16:10,600
But we know,
actually, how to find

348
00:16:10,600 --> 00:16:12,290
the area of this component.

349
00:16:12,290 --> 00:16:12,790
OK.

350
00:16:12,790 --> 00:16:14,069
So hopefully this makes sense.

351
00:16:14,069 --> 00:16:16,110
Because let's think about,
when I'm finding area,

352
00:16:16,110 --> 00:16:18,050
I'm going from the
origin and I'm coming out

353
00:16:18,050 --> 00:16:20,020
and I have the radius out there.

354
00:16:20,020 --> 00:16:24,770
So when I integrate this
dA from 0 to pi for theta,

355
00:16:24,770 --> 00:16:27,630
I'm picking up
pieces that come out,

356
00:16:27,630 --> 00:16:30,500
little sectors that
come out like this

357
00:16:30,500 --> 00:16:31,880
between pi over 2 and pi.

358
00:16:31,880 --> 00:16:34,310
So I'm getting more area
than I want if I just

359
00:16:34,310 --> 00:16:36,740
let theta go between 0 and pi.

360
00:16:36,740 --> 00:16:38,900
So I have to
calculate all of it,

361
00:16:38,900 --> 00:16:41,751
and then I have to take
away the extra stuff.

362
00:16:41,751 --> 00:16:42,250
OK.

363
00:16:42,250 --> 00:16:44,820
So the blue area is
actually the bigger area

364
00:16:44,820 --> 00:16:47,180
subtracting the smaller area.

365
00:16:47,180 --> 00:16:48,890
And so how am I
going to write this?

366
00:16:48,890 --> 00:16:51,230
If we come back over, I'm
just going to take 2 times

367
00:16:51,230 --> 00:16:52,440
this whole thing.

368
00:16:52,440 --> 00:16:54,220
So I'm going to
take 2 times this.

369
00:16:54,220 --> 00:16:57,580
And I know I have to
integrate it from 0 to pi.

370
00:16:57,580 --> 00:17:00,850
And I'm taking 2 times because
it's symmetric, remember.

371
00:17:00,850 --> 00:17:04,310
And then I'm going to subtract
track off this one that's 2

372
00:17:04,310 --> 00:17:06,250
times the thing from pi to 2*pi.

373
00:17:06,250 --> 00:17:07,750
Now, you might say,
well, Christine,

374
00:17:07,750 --> 00:17:12,190
the pink stuff I'm interested
in between 0 and pi is actually

375
00:17:12,190 --> 00:17:15,714
theta between not pi and
2*pi, but 2*pi and 3*pi.

376
00:17:15,714 --> 00:17:17,880
But again, there's all this
symmetry in the problem.

377
00:17:17,880 --> 00:17:20,270
So it doesn't really matter.

378
00:17:20,270 --> 00:17:22,310
But if you're a
stickler, I guess

379
00:17:22,310 --> 00:17:24,790
I'll even write it this
way just to make sure.

380
00:17:24,790 --> 00:17:27,869
So I'll write it as 2*pi to 3*pi
so that everyone's very happy.

381
00:17:27,869 --> 00:17:28,369
OK?

382
00:17:34,920 --> 00:17:38,240
So again, what do we do?

383
00:17:38,240 --> 00:17:39,460
These 2's divide out.

384
00:17:39,460 --> 00:17:42,600
So from 0 to pi of
r squared d theta,

385
00:17:42,600 --> 00:17:46,890
that's going to give me the
area of the blue plus the pink.

386
00:17:46,890 --> 00:17:51,010
And then 2*pi to 3*pi of 1
plus cosine over theta over 2

387
00:17:51,010 --> 00:17:54,510
squared d theta is going to
give me the area of the pink.

388
00:17:54,510 --> 00:17:57,920
So the blue plus the
pink is over here,

389
00:17:57,920 --> 00:17:59,320
and then the pink is over here.

390
00:17:59,320 --> 00:18:02,280
So when I subtract that
off I just get the blue.

391
00:18:02,280 --> 00:18:03,030
OK.

392
00:18:03,030 --> 00:18:05,562
Let me, again, just
go back one more time

393
00:18:05,562 --> 00:18:07,520
and point out what we
did at the very beginning

394
00:18:07,520 --> 00:18:10,540
to remind us what was happening.

395
00:18:10,540 --> 00:18:11,540
And then I will finish.

396
00:18:11,540 --> 00:18:13,700
So let's come back over here.

397
00:18:13,700 --> 00:18:17,056
So the idea was to graph
this curve that was

398
00:18:17,056 --> 00:18:19,620
in r is a function of theta.

399
00:18:19,620 --> 00:18:21,410
And I was supposed to
understand what that

400
00:18:21,410 --> 00:18:24,140
looked like in the xy-plane.

401
00:18:24,140 --> 00:18:30,280
And so my trick was to take
the relationship between r

402
00:18:30,280 --> 00:18:34,020
and theta and graph
that explicitly in an r,

403
00:18:34,020 --> 00:18:35,996
theta plane-- so
I let theta vary

404
00:18:35,996 --> 00:18:37,370
in the horizontal
direction and r

405
00:18:37,370 --> 00:18:41,160
vary in the vertical direction--
and I can do that very easily.

406
00:18:41,160 --> 00:18:44,830
And then translate
that into the xy-plane.

407
00:18:44,830 --> 00:18:48,730
So my curve, again,
went the big part,

408
00:18:48,730 --> 00:18:52,390
the little part here, little
part here, big part here.

409
00:18:52,390 --> 00:18:53,760
That was the order.

410
00:18:53,760 --> 00:18:58,045
So if you need arrows on
it, this was the order.

411
00:19:00,820 --> 00:19:04,890
And then once I had that,
the problem was about areas.

412
00:19:04,890 --> 00:19:08,200
And there was a lot of
symmetry in this problem,

413
00:19:08,200 --> 00:19:10,260
but the main point
I wanted to show

414
00:19:10,260 --> 00:19:15,640
was just knowing where your
theta starts and stops is not

415
00:19:15,640 --> 00:19:17,940
enough to determine
an area of a region,

416
00:19:17,940 --> 00:19:21,430
if that region is excluding some
part that would be potentially

417
00:19:21,430 --> 00:19:22,780
counted twice.

418
00:19:22,780 --> 00:19:25,940
So that was the reason I wanted
you to calculate not just

419
00:19:25,940 --> 00:19:30,420
the pink area, but also see
that the blue is not from theta

420
00:19:30,420 --> 00:19:34,250
from 0 to pi, but you have to
subtract off this extra stuff

421
00:19:34,250 --> 00:19:36,230
that you counted.

422
00:19:36,230 --> 00:19:38,946
And I guess that is
where I will stop.