The following function: $$\iiint_{\mathrm{E}}(x^2+xy+y^2)\, dx\,dy\,dz, $$ where $E$ is a globe with radius 2 and center in origin.
I have got the following limits using spherical coordinates $0 \le r \le 2$, $0 \le \theta \le \pi$ and $0 \le \phi \le 2\pi$. I'm unsure if I've calculated correctly.
Due to symmetry $$\iiint_{\mathrm{E}}xy\, dx\,dy\,dz=0;\>\>\> \iiint_{\mathrm{E}}x^2\, dx\,dy\,dz =\iiint_{\mathrm{E}}y^2\, dx\,dy\,dz =\iiint_{\mathrm{E}}z^2\, dx\,dy\,dz =I$$ Thus $$\iiint_{\mathrm{E}}(x^2+xy+y^2)\, dx\,dy\,dz =\frac23(3I)=\frac23\iiint_{\mathrm{E}}(x^2+y^2+z^2)\, dx\,dy\,dz\\ = \frac23 \int_0^{2\pi}\int_0^{\pi}\int_0^2r^2\>r^2\sin\theta drd\theta d\phi=\frac23\cdot2\pi\cdot2\cdot\frac{32}5=\frac{256}{15}\pi $$