Area with Polar Coordinates — Question 1

Question 1 – 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. $A=\tfrac12\int_0^{2\pi}(1+\cos\theta)^2d\theta=\boxed{3\pi/2}.$