Question

find the area of the region that lies inside the cardiod r=1+cos(theta) and the outside of the circle r=1

Answer 1

\[\int\limits_{0}^{2\pi}1+\cos(\theta) d \theta - \pi\]
You're just taking the outer cardiod and subtracting the area of the circle inside. For the first part, you're evaluating around theta 0 to 2pi because that's a full circle, and then the second part the area of a circle with r=1 is pi.

Answer 2

thank you!

Answer 3

This is of course assuming that at no point does the perimeter of the circle intersect that of the cardiod. You should check that.