Page 58 - Calculus for the Clueless

Find the area of the region bounded by the cardioid r = 2 + 2 cos θ (inside this curve) and the circle r = 3
(outside this curve).

See the figure.

Since the area is inside the cardioid, the cardioid is the outside curve. Outside r = 3 makes the circle the inside
curve. In order to find the limits of the integral, we set the r's equal to each other; 3 = 2 + 2 cos θ; cos θ=½;
θ=±π/3. Also, there is symmetry about the x-axis. So we can double the integral from 0 to π/3.

There are two more problems that we can illustrate with the same diagram.

Example 13—

Find the area outside the cardioid and inside the circle.

If it's inside the circle r = 3, r = 3 becomes the outside r. ''Outside the cardioid" means r = 2 + 2 cos θ becomes
the inside curve. Again, we have x-axis symmetry. Again, setting the curves equal, we get ±π/3. The picture
tells us twice the integral from π/3 to π. Since we did the last problem, we need not integrate twice, since it is
the same integral. We need only change all the signs.