Area with Polar Coordinates — Question 2

Question 2

Problem

Find the area of one petal of $r=3\cos2\theta$.

Question 2 – 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. One petal is traced on $[-\pi/4,\pi/4]$.

4. Thus $A=\tfrac12\int_{-\pi/4}^{\pi/4}9\cos^22\theta d\theta=\boxed{9\pi/8}.$