Area with Polar Coordinates — Question 3

Question 3

Problem

Find the area of the inner loop of $r=1-2\cos\theta$.

Question 3 – 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 loop is traced for $-\pi/3\le\theta\le\pi/3$.

4. Hence $A=\tfrac12\int_{-\pi/3}^{\pi/3}(1-2\cos\theta)^2d\theta=\boxed{\pi-3\sqrt3/2}.$