Provide an Appropriate Response $r ^ { 2 } = a ^ { 2 } \cos 2 \theta$

Essay

Provide an appropriate response.
-Find the error in the following "proof" that the area inside the lemniscate $r ^ { 2 } = a ^ { 2 } \cos 2 \theta$ is 0 and then find the correct area.
$\left. A = \int _ { 0 } ^ { 2 \pi } \frac { 1 } { 2 } r ^ { 2 } d \theta = \int _ { 0 } ^ { 2 \pi } \frac { 1 } { 2 } a ^ { 2 } \cos 2 \theta d \theta = \frac { 1 } { 4 } a ^ { 2 } \sin 2 \theta \right] _ { 0 } ^ { 2 \pi } = 0$