Area with Polar Coordinates — Question 8

Question 8

Problem

Two polar curves satisfy $0\le r_1(\theta)\le r_2(\theta)$. Explain why the area between them involves squares, not merely $r_2-r_1$.

Question 8 – 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. A thin polar sector has area $\tfrac12r^2d\theta$.

4. Subtracting sector areas yields $\tfrac12(r_2^2-r_1^2)d\theta$, reflecting increasing width with radius.