AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.9 Finding the Area of the Region Bounded by Two Polar Curves
Notes
$$A=\frac{1}{2}\int_\alpha^\beta\left(r_{outer}^2-r_{inner}^2\right)d\theta$$
- Find where the curves meet by setting the $r$'s equal.
Practice
$r=2$ and $r=4\cos\theta$ meet where $\cos\theta=$
$4\cos\theta=2$.
Area inside $r=2$ and outside $r=1$:
$\displaystyle \frac{1}{2}\int_0^{2\pi}(4-1)d\theta$.
For the area between two polar curves, you subtract:
Square each, then subtract.
Ready for more? Take the Unit 9 test.