Area with Polar Coordinates — Question 10

Question 10 – Solution

Solution

1. Determine an interval of angles that sweeps out the requested region exactly once, using symmetry when it simplifies the work.

2. For a polar boundary $r=f(\theta)$, use $$A=\frac12\int_a^b r^2\,d\theta.$$ For a region between two curves, subtract the inner squared radius from the outer squared radius before integrating.

3. It is a circle of radius $a/2$, so area is $\pi a^2/4$.

4. Setting this to $9\pi$ gives $\boxed{a=6}$.