I'm asked to integrate $\vec{A}=r\cos\theta \ \vec{u_r}+r\sin\theta\ \vec{u_\theta}+r\sin\theta\cos\varphi \ \vec{u_\varphi}$ over a hemisphere of radius $a$.
I take $d\vec{S}=r^2\sin\theta d\theta d\varphi \ \vec{u_r}$, and therefore:
$$\iint_S\vec{A}\ \cdot d\vec{S}=\int_{\theta=0}^{\pi/2}\int_{\varphi=0}^{2\pi}a^3\sin\theta \cos\theta \ d\theta d\varphi=2\pi a^3 \int_0^{2\pi}\sin\theta\cos\theta \ d\theta=\pi a^3\int_{0}^{2\pi}\sin(2\theta) \ d\theta=\frac{\pi a^3}{2}[-\cos(\pi)+\cos(0)]=\pi a^3$$
However the results state that it should be $5\pi a^3 /3$. So where am I wrong? Or are the results incorrect?