WEBVTT

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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,

00:09:54.540 --> 00:09:56.210
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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Yeah, there is
positive radius there.

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

00:10:06.540 --> 00:10:08.910
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.

00:10:11.850 --> 00:10:15.650
So then what happens
between 2*pi and 3*pi?

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OK.

00:10:16.150 --> 00:10:18.310
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.

00:10:20.740 --> 00:10:22.390
What happens between
2*pi and 3*pi?

00:10:22.390 --> 00:10:24.020
2*pi, we're here.

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

00:10:26.720 --> 00:10:30.690
And notice that the radius is
going to be doing the same kind

00:10:30.690 --> 00:10:35.350
of growth that it did between
2*pi and 3*pi as it did decay

00:10:35.350 --> 00:10:36.845
from pi to 2*pi.

00:10:36.845 --> 00:10:37.500
OK?

00:10:37.500 --> 00:10:39.910
Because the radii now,
there's a symmetry

00:10:39.910 --> 00:10:41.490
with how the radii behave.

00:10:41.490 --> 00:10:46.600
So from 2*pi to 3*pi, I start
off with radius 0 and I have

00:10:46.600 --> 00:10:48.120
a small radius.

00:10:48.120 --> 00:10:53.380
And then once I get
to 3*pi over here,

00:10:53.380 --> 00:10:54.756
I'm going to have
radius 1 again.

00:10:54.756 --> 00:10:56.546
I'm going to be at
radius 1, which is going

00:10:56.546 --> 00:10:57.840
to correspond to this point.

00:10:57.840 --> 00:11:00.390
So I'm going to have exactly
the same picture, which

00:11:00.390 --> 00:11:03.265
is dangerous because I
probably would do it wrong.

00:11:03.265 --> 00:11:06.840
But I'll try and draw it this
way and then talk about it.

00:11:06.840 --> 00:11:09.480
Hopefully that looks
about the same.

00:11:09.480 --> 00:11:13.570
So this curve coming through
here was from pi to 2*pi.

00:11:13.570 --> 00:11:17.750
This curve coming through
here was from to 2*pi to 3*pi.

00:11:17.750 --> 00:11:20.470
And then to finish it off, 3*pi
to 4*pi is going to look like

00:11:20.470 --> 00:11:22.310
this curve here.

00:11:22.310 --> 00:11:26.150
So I come through-- ooh, this
is where it starts to get really

00:11:26.150 --> 00:11:29.980
dangerous, but let's say
that's pretty close--

00:11:29.980 --> 00:11:33.160
so there's my 3*pi to 4*pi.

00:11:33.160 --> 00:11:35.660
It's again, the
same growth the way

00:11:35.660 --> 00:11:38.260
it was decaying
between 0 and pi.

00:11:38.260 --> 00:11:44.360
So this is your picture in the
xy-plane of the curve r equals

00:11:44.360 --> 00:11:47.150
1 plus cosine theta over 2.

00:11:47.150 --> 00:11:50.900
Now, we haven't calculated
any area problems, yet.

00:11:50.900 --> 00:11:55.470
So what I'd like us to do is
I'm going to shade two regions,

00:11:55.470 --> 00:12:00.430
and I want us to just write
down the integral form

00:12:00.430 --> 00:12:04.070
to find the area for
each of these regions.

00:12:04.070 --> 00:12:09.600
So the first region of
interest is the pink region.

00:12:09.600 --> 00:12:12.750
I'm going to ask us to find
the area of the pink region,

00:12:12.750 --> 00:12:16.420
and then I'm going to ask us
to find the area of everything

00:12:16.420 --> 00:12:17.435
else, the blue region.

00:12:20.450 --> 00:12:24.480
So let's think about
how to do that.

00:12:24.480 --> 00:12:26.980
And I think I'm going to have
to come over to the other side

00:12:26.980 --> 00:12:30.440
to write down the integrals, but
I'll be coming back and forth.

00:12:30.440 --> 00:12:32.660
So just to remind
you, what you saw

00:12:32.660 --> 00:12:40.550
in lecture was that dA is equal
to r squared over 2 d theta.

00:12:40.550 --> 00:12:42.740
That's what dA is.

00:12:42.740 --> 00:12:45.960
And so this is going to
be an integral in theta,

00:12:45.960 --> 00:12:48.650
and we know what r is
as a function of theta.

00:12:48.650 --> 00:12:50.150
And so if I want
to find-- actually,

00:12:50.150 --> 00:12:54.330
I should even use my
colors appropriately.

00:12:54.330 --> 00:12:59.710
I should say, the
pink area is going

00:12:59.710 --> 00:13:01.860
to be equal to an integral.

00:13:01.860 --> 00:13:03.780
And I'm going to
write the r squared.

00:13:03.780 --> 00:13:05.010
I know what r squared is.

00:13:05.010 --> 00:13:10.200
It's 1 plus cosine
theta over 2 squared,

00:13:10.200 --> 00:13:12.560
and then an over 2 d theta.

00:13:12.560 --> 00:13:14.810
And then what is
important about this?

00:13:14.810 --> 00:13:15.590
It's our bounds.

00:13:15.590 --> 00:13:16.090
Right?

00:13:16.090 --> 00:13:17.800
Our bounds are what's important.

00:13:17.800 --> 00:13:21.760
And so let's go back and
let's look at our picture.

00:13:21.760 --> 00:13:25.680
What are the bounds on
theta for the pink region?

00:13:25.680 --> 00:13:28.540
So where does that pink
region start and stop?

00:13:28.540 --> 00:13:32.530
And maybe we even need to
look at this top graph, also.

00:13:32.530 --> 00:13:36.280
So if we think about
it, we went from 0 to pi

00:13:36.280 --> 00:13:38.066
to get this outer curve.

00:13:38.066 --> 00:13:39.440
So how do we get
the inner curve?

00:13:39.440 --> 00:13:44.070
The inner curve started at
theta equals pi, went to here,

00:13:44.070 --> 00:13:47.900
went to here-- that
was theta equals 3*pi.

00:13:47.900 --> 00:13:52.160
So it went all the
way from pi to 3*pi.

00:13:52.160 --> 00:13:54.780
Now, if you're paying
good attention,

00:13:54.780 --> 00:13:58.300
you can say, well, Christine,
we know that this region is

00:13:58.300 --> 00:13:59.550
totally symmetric.

00:13:59.550 --> 00:14:02.730
So why don't I just take
the area from pi to 2*pi

00:14:02.730 --> 00:14:04.470
and multiply it by 2?

00:14:04.470 --> 00:14:05.340
And you can.

00:14:05.340 --> 00:14:06.910
You can do it either way.

00:14:06.910 --> 00:14:09.900
So you can either take the
integral from all the way from

00:14:09.900 --> 00:14:13.880
pi to 3*pi, which corresponds
to starting at this angle,

00:14:13.880 --> 00:14:16.324
going all the way around,
and coming back to there,

00:14:16.324 --> 00:14:18.240
which takes you all the
way around this curve.

00:14:18.240 --> 00:14:21.780
Or you can go from pi to
2*pi and multiply that by 2.

00:14:21.780 --> 00:14:25.470
So let me come back and
let me write that down.

00:14:25.470 --> 00:14:32.430
It's either pi to 3*pi, this,
or you just write it as integral

00:14:32.430 --> 00:14:34.310
from pi to 2*pi.

00:14:34.310 --> 00:14:38.430
And if I multiply this
by 2, the 2 drops out.

00:14:38.430 --> 00:14:40.477
The 2 in the
denominator drops out.

00:14:44.407 --> 00:14:46.240
I'm not going to solve
this problem for you,

00:14:46.240 --> 00:14:48.781
but I do want to point out the
kinds of terms you would have.

00:14:48.781 --> 00:14:51.060
You would have a constant
term when you square this.

00:14:51.060 --> 00:14:54.150
You would have a term that
was 2 cosine theta over 2,

00:14:54.150 --> 00:14:56.980
which is easy to integrate
by a u-substitution.

00:14:56.980 --> 00:14:59.450
And then you would have
a cosine squared theta

00:14:59.450 --> 00:15:02.460
over 2, which you'd want
to use the double angle

00:15:02.460 --> 00:15:03.940
formula or the
half angle formula

00:15:03.940 --> 00:15:07.060
that you've seen used to
manipulate these integrals that

00:15:07.060 --> 00:15:09.680
involve just a cosine
squared or a sine squared.

00:15:09.680 --> 00:15:11.600
So that would be your strategy.

00:15:11.600 --> 00:15:13.175
OK, now let's look
at the blue area.

00:15:13.175 --> 00:15:13.675
OK.

00:15:19.090 --> 00:15:20.840
So to find the
blue area, again, I

00:15:20.840 --> 00:15:24.307
know all that matters
is really the bounds.

00:15:24.307 --> 00:15:26.515
We're going to see we have
to do a little extra work.

00:15:29.600 --> 00:15:31.000
But this is our first setup.

00:15:31.000 --> 00:15:32.861
And now let's go
look at the bounds.

00:15:32.861 --> 00:15:33.360
OK.

00:15:33.360 --> 00:15:35.880
So we go back to the curve.

00:15:35.880 --> 00:15:37.020
All right.

00:15:37.020 --> 00:15:38.000
What is the blue area?

00:15:38.000 --> 00:15:41.730
Well, the blue area,
that's a little harder.

00:15:41.730 --> 00:15:43.680
So let's see what happens.

00:15:43.680 --> 00:15:49.490
If I were to take theta from
0 to pi, what would happen?

00:15:49.490 --> 00:15:51.460
I would not only pick
up the blue area,

00:15:51.460 --> 00:15:54.100
but I'd pick up this
pink stuff inside.

00:15:54.100 --> 00:15:55.640
But I don't want the pink stuff.

00:15:55.640 --> 00:15:57.290
I just want the blue stuff.

00:15:57.290 --> 00:16:00.020
So what am I going to have to
do to find the blue area just,

00:16:00.020 --> 00:16:01.980
say, from 0 up to pi?

00:16:01.980 --> 00:16:04.641
I'm going to have to find
the area from 0 to pi,

00:16:04.641 --> 00:16:06.390
and then I'm going to
have to subtract off

00:16:06.390 --> 00:16:07.722
the area of this component.

00:16:07.722 --> 00:16:09.184
OK?

00:16:09.184 --> 00:16:10.600
But we know,
actually, how to find

00:16:10.600 --> 00:16:12.290
the area of this component.

00:16:12.290 --> 00:16:12.790
OK.

00:16:12.790 --> 00:16:14.069
So hopefully this makes sense.

00:16:14.069 --> 00:16:16.110
Because let's think about,
when I'm finding area,

00:16:16.110 --> 00:16:18.050
I'm going from the
origin and I'm coming out

00:16:18.050 --> 00:16:20.020
and I have the radius out there.

00:16:20.020 --> 00:16:24.770
So when I integrate this
dA from 0 to pi for theta,

00:16:24.770 --> 00:16:27.630
I'm picking up
pieces that come out,

00:16:27.630 --> 00:16:30.500
little sectors that
come out like this

00:16:30.500 --> 00:16:31.880
between pi over 2 and pi.

00:16:31.880 --> 00:16:34.310
So I'm getting more area
than I want if I just

00:16:34.310 --> 00:16:36.740
let theta go between 0 and pi.

00:16:36.740 --> 00:16:38.900
So I have to
calculate all of it,

00:16:38.900 --> 00:16:41.751
and then I have to take
away the extra stuff.

00:16:41.751 --> 00:16:42.250
OK.

00:16:42.250 --> 00:16:44.820
So the blue area is
actually the bigger area

00:16:44.820 --> 00:16:47.180
subtracting the smaller area.

00:16:47.180 --> 00:16:48.890
And so how am I
going to write this?

00:16:48.890 --> 00:16:51.230
If we come back over, I'm
just going to take 2 times

00:16:51.230 --> 00:16:52.440
this whole thing.

00:16:52.440 --> 00:16:54.220
So I'm going to
take 2 times this.

00:16:54.220 --> 00:16:57.580
And I know I have to
integrate it from 0 to pi.

00:16:57.580 --> 00:17:00.850
And I'm taking 2 times because
it's symmetric, remember.

00:17:00.850 --> 00:17:04.310
And then I'm going to subtract
track off this one that's 2

00:17:04.310 --> 00:17:06.250
times the thing from pi to 2*pi.

00:17:06.250 --> 00:17:07.750
Now, you might say,
well, Christine,

00:17:07.750 --> 00:17:12.190
the pink stuff I'm interested
in between 0 and pi is actually

00:17:12.190 --> 00:17:15.714
theta between not pi and
2*pi, but 2*pi and 3*pi.

00:17:15.714 --> 00:17:17.880
But again, there's all this
symmetry in the problem.

00:17:17.880 --> 00:17:20.270
So it doesn't really matter.

00:17:20.270 --> 00:17:22.310
But if you're a
stickler, I guess

00:17:22.310 --> 00:17:24.790
I'll even write it this
way just to make sure.

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.

00:17:27.869 --> 00:17:28.369
OK?

00:17:34.920 --> 00:17:38.240
So again, what do we do?

00:17:38.240 --> 00:17:39.460
These 2's divide out.

00:17:39.460 --> 00:17:42.600
So from 0 to pi of
r squared d theta,

00:17:42.600 --> 00:17:46.890
that's going to give me the
area of the blue plus the pink.

00:17:46.890 --> 00:17:51.010
And then 2*pi to 3*pi of 1
plus cosine over theta over 2

00:17:51.010 --> 00:17:54.510
squared d theta is going to
give me the area of the pink.

00:17:54.510 --> 00:17:57.920
So the blue plus the
pink is over here,

00:17:57.920 --> 00:17:59.320
and then the pink is over here.

00:17:59.320 --> 00:18:02.280
So when I subtract that
off I just get the blue.

00:18:02.280 --> 00:18:03.030
OK.

00:18:03.030 --> 00:18:05.562
Let me, again, just
go back one more time

00:18:05.562 --> 00:18:07.520
and point out what we
did at the very beginning

00:18:07.520 --> 00:18:10.540
to remind us what was happening.

00:18:10.540 --> 00:18:11.540
And then I will finish.

00:18:11.540 --> 00:18:13.700
So let's come back over here.

00:18:13.700 --> 00:18:17.056
So the idea was to graph
this curve that was

00:18:17.056 --> 00:18:19.620
in r is a function of theta.

00:18:19.620 --> 00:18:21.410
And I was supposed to
understand what that

00:18:21.410 --> 00:18:24.140
looked like in the xy-plane.

00:18:24.140 --> 00:18:30.280
And so my trick was to take
the relationship between r

00:18:30.280 --> 00:18:34.020
and theta and graph
that explicitly in an r,

00:18:34.020 --> 00:18:35.996
theta plane-- so
I let theta vary

00:18:35.996 --> 00:18:37.370
in the horizontal
direction and r

00:18:37.370 --> 00:18:41.160
vary in the vertical direction--
and I can do that very easily.

00:18:41.160 --> 00:18:44.830
And then translate
that into the xy-plane.

00:18:44.830 --> 00:18:48.730
So my curve, again,
went the big part,

00:18:48.730 --> 00:18:52.390
the little part here, little
part here, big part here.

00:18:52.390 --> 00:18:53.760
That was the order.

00:18:53.760 --> 00:18:58.045
So if you need arrows on
it, this was the order.

00:19:00.820 --> 00:19:04.890
And then once I had that,
the problem was about areas.

00:19:04.890 --> 00:19:08.200
And there was a lot of
symmetry in this problem,

00:19:08.200 --> 00:19:10.260
but the main point
I wanted to show

00:19:10.260 --> 00:19:15.640
was just knowing where your
theta starts and stops is not

00:19:15.640 --> 00:19:17.940
enough to determine
an area of a region,

00:19:17.940 --> 00:19:21.430
if that region is excluding some
part that would be potentially

00:19:21.430 --> 00:19:22.780
counted twice.

00:19:22.780 --> 00:19:25.940
So that was the reason I wanted
you to calculate not just

00:19:25.940 --> 00:19:30.420
the pink area, but also see
that the blue is not from theta

00:19:30.420 --> 00:19:34.250
from 0 to pi, but you have to
subtract off this extra stuff

00:19:34.250 --> 00:19:36.230
that you counted.

00:19:36.230 --> 00:19:38.946
And I guess that is
where I will stop.