Area with Polar Coordinates — Question 5

Question 5

Problem

Find the area inside $r=2$ but outside $r=2\cos\theta$.

Question 5 – 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. The circle $r=2\cos\theta$ lies entirely inside $r=2$.

4. Subtract areas: $4\pi-\pi=\boxed{3\pi}.$