Finding are of a region that lies inside the first curve and outside the second for $r=5-5sin(\theta) $ and $r=5$

Can someone please help me figure out where I'm messing up in this problem?

1 Answer

Your calculation is basically fine. However, as part of the evaluation, we need $$\left.\frac{25\theta}{4}\right|_{\pi}^{2\pi}.$$ This is $\frac{25\pi}{4}$, not $25\pi$.

Remark: The "$50$" term is right, but may have been reached by two cancelling errors. In the evaluation, you seem to be adding the values at $2\pi$ and $\pi$, not subtracting. Note also that $\cos(\pi)=-1$.