Area with Polar Coordinates — Question 4

Question 4

Problem

Find the common area of the circles $r=2\cos\theta$ and $r=2\sin\theta$.

Question 4 – 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. In the first quadrant the smaller radius switches at $\pi/4$.

4. Symmetry gives $A=\int_0^{\pi/4}(2\sin\theta)^2d\theta=\boxed{\pi/2-1}.$