Question

please
express as a double integraj\l the area enclosed by one loop of the curve r=3cos2\theta

Answer 1

\(r=3\cos2\theta\) or \(r=3\cos^2\theta\)?

Answer 2

r=3\cos2\theta

Answer 3

Here's a sketch of one loop:

This region can be defined by the set,
\[R:=\left\{(r,\theta)~:~-\frac{\pi}{4}\le\theta\le\frac{\pi}{4},~0\le r\le 3\cos2\theta\right\}\]
As a double integral, the area is then
\[\large A=\int_{-\pi/4}^{\pi/4}\int_0^{3\cos2\theta}~r~dr~d\theta\]
Using symmetry, you can evaluate it as
\[\large A=2\int_0^{\pi/4}\int_0^{3\cos2\theta}~r~dr~d\theta\]