Area with Polar Coordinates — Question 6

Question 6

Problem

The curve $r^2=4\cos2\theta$ has two loops. Find its total area.

Question 6 – 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. One loop uses $-\pi/4\le\theta\le\pi/4$: area $=\tfrac12\int4\cos2\theta d\theta=2$.

4. Doubling gives $\boxed{4}$.