Area with Polar Coordinates — Question 9

Question 9

Problem

Find the area enclosed by one loop of $r=2\sin3\theta$ and explain why multiplying by six is wrong for total area.

Question 9 – 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 loop area is $\tfrac12\int_0^{\pi/3}4\sin^23\theta d\theta=\pi/3$.

4. The rose has three petals, not six, because negative radii retrace opposite petals; total area is $\boxed{\pi}$.