Concept preview

What you will be able to do

1. 01Set up the area of a region bounded by two polar curves as $A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta \left(r_{\text{outer}}^2-r_{\text{inner}}^2\right)\,d\theta$, subtracting the SQUARES of the radii — never the square of the difference.
2. 02Find the correct angle limits $\alpha$ and $\beta$ by locating the intersection(s) of the two curves, that is, by solving $r_1=r_2$, and decide over which $\theta$-interval one curve is the outer boundary and the other the inner boundary.
3. 03Evaluate the resulting integral exactly for classic configurations — a region inside one curve and outside another, or the overlap of a circle with a cardioid or a second circle — keeping $\pi$ in symbolic form.

Start from the one thing you already trust. A region swept by a single polar curve $r(\theta)$ between the rays $\theta=\alpha$ and $\theta=\beta$ has area $A=\dfrac{1}{2}\displaystyle\int_\alpha^\beta r^2\,d\theta$, because each infinitesimal wedge is a circular sector of central angle $d\theta$ and radius $r$, whose area is $\tfrac{1}{2}r^2\,d\theta$. Hold onto the picture of a wedge, because the two-curve case is built entirely out of it.