Area with Polar Coordinates — Question 7

Question 7

Problem

A radar beam reaches distance $r=5+2\sin\theta$. Find the area swept while $0\le\theta\le\pi$.

Question 7 – 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. $A=\tfrac12\int_0^\pi(5+2\sin\theta)^2d\theta=\boxed{27\pi/2+20}.$