I'm told that the area enclosed by the curve $r=a+5\sin\theta$, $a>5$ is given by $187\pi/2$, and I'm then asked to find the value of $a$.
I know that the area enclosed by a polar curve is given by $$\text{Area}=\frac{1}{2}\int_{\theta_1}^{\theta_2}r^2 \, d\theta$$
If I use this with $\theta_1=0$ and $\theta_2=2\pi$, I find that $a=9$, which I'm told is the correct answer.
However, if I use the symmetry of this curve, integrating between $0$ and $\pi$ and then doubling my answer, I find values of $a$ such that $a\neq 9$.
Why is this?
The curve is not symmetric about $\theta=0$ (a horizontal line). It is symmetric about $\theta=\frac\pi2$ (a vertical line), so if you want to exploit symmetry your reduced bounds should run from $-\frac\pi2$ to $\frac\pi2$.