Can't find the area of a polar region

Question

I've ran into a bit of a stopper on this one problem. I solved this other problem like this yesterday but this one seems to cancel itself out to zero. I'm not sure what I'm doing wrong with this problem.

the original equation is: r^2 = 81 cos(2x)

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this is what I've done to solve it so far

My work

and I know that the graph looks like this

My graph

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The inetgral keeps canceling out to be zero and I'm not sure what to do with this andy more.

Answer 1

Of course it is zero, because you are summing over a whole period of $\cos 2\theta$.

I suggest you calculate the area of one petal and multiply it by $8$. That is, $$ 8 \cdot \frac{81}{2}\int_{-\pi/4}^{\pi/4} \cos 2\theta\,d\theta $$ The reason to do so is sometimes the radius is negative, but the graph still encloses positive area.

Edit: The graph only has two petals, so the above needs to be changed.