I can't seem to work out this problem:
Find the flux of $F(x,y,z) = 3xy^2i + 3x^2yj +z^3k$ out of the unit sphere centred at (0,0,0).
My attempt is as follows:
\begin{align*} \iint_S F \cdot dS &= \iint_S [F(x,y,z) \cdot \hat{n}]dS\\ &\text{Since unit sphere:}\\ &=\iint_S [(3xy^2,3x^2y,z^3) \cdot (x,y,z)]dS && \\ &=\iint_S [3x^2y^2 + 3x^2y^2 + z^4 ]dS\\ &=\iint_S [6x^2y^2 + z^4 ] dS\\ &\text{ Converstion to spherical coordinates, with } r=1:\\ &=\int_0^{2\pi} \int_0^{\pi} [(6(sin(\theta) cos(\phi))^2(sin(\theta) sin(\phi))^2 + cos^4(\theta)) \cdot sin(\theta)] d\theta d\phi\\ &=\int_0^{2\pi} \int_0^{\pi} [(6sin^4(\theta)cos^2(\phi)sin^2(\phi) + cos^4(\theta)) \cdot sin(\theta)] d\theta d\phi\\ \end{align*}
At this point I can't work out how to simplify it further in order to do the integration. Is there a mistake somewhere or another way to proceed?