I've managed to calculate this integral
$$\intop_{D}\left(2x^{2}y-6x^{2}y^{2}+2x^{2}yz\right)dx\,dy\,dz$$
where $D$ is given by $x^{2}+y^{2}+z^{2}\leq1,$ by following the author's advice and converting it to spherical coordinates. However, it was extremely labourious and I'm wondering if I've missed any shortcuts. Could anyone please advise?
The initial conversion to spherical coordinates gave
$$\int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{1}\left(\left(2r^{5}\sin^{4}\phi\cos^{2}\theta\sin\theta\right)-\left(6r^{6}\sin^{5}\phi\cos^{2}\theta\sin^{2}\theta\right)+\left(2r^{6}\sin^{4}\phi\cos^{2}\theta\sin\theta\cos\phi\right)\right)dr\,d\theta\,d\phi$$ and eventually I reached
$$\int_{0}^{\pi}\left(0-\left(3/14\right)\pi\sin^{5}\phi+0\right)d\phi=\frac{-8\pi}{35}.$$Is there a simpler method?