18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 33
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PROFESSOR: So again, welcome back. And today's topic is a continuation of what we
did last time. We still have a little bit of work and thinking to do concerning polar
coordinates. So we're going to talk about polar coordinates. And my first job today is
to talk a little bit about area. That's something we didn't mention last time. And since
we're all back from Thanksgiving, we can certainly talk about it in terms of a pie.
Which is the basic idea for area in polar coordinates. Here's our pie, and here's a
slice of the pie. The slice has a piece of arc length on it, which I'm going to call delta
theta. And the area of that shaded-in slice, I'm going to call delta A.
And let's suppose that the radius is a. Little a. So this is a pie of radius a. That's our
picture. Now, it's pretty easy to figure out what the area that slice of pie is. The total
area is, of course, pi a ^2. We know that. And to get this fraction, delta A, all we
have to do is take the percentage of the arc of the total circumference. That's delta
theta / 2 pi. This is the fraction of area -- sorry, fraction of the total circumference,
the total length around the rim. And then we multiply that by pi a ^2. And that's
giving us the total area. And if you work that out, that's delta A is equal to, the pi's
cancel and we have 1/2 a ^2 delta theta.
So here's the basic formula. And now what we need to do is to talk about a variable
pie here. That would be a pie with a kind of a wavy crust. Which is coming around
like this. So r = r (theta). The distance from the center is varying with the place
where we are, the angle where we're shooting out. And now I want to subdivide that
into little chunks here. Now, the idea for adding up the area, the total area of this
piece that's swept out, is to break it up into little slices whose areas are almost easy
to calculate. Namely, what we're going to do is to take, and I'm going to label it this
way. I'm going to take these little circular arcs, which go. So I'm going to extend
past where this goes. And then I'm going to take each circular arc here. So here's a