Question

E xex2 + y2 + z2 dv, where e is the portion of the unit ball x2 + y2 + z2 ≤ 1 that lies in the first octant.

Answer 1

$\displaystyle\iiint_{\mathcal E}xe^{x^2+y^2+z^2}\,\mathrm dV$
$=\displaystyle\int_{\varphi=0}^{\varphi=\pi/2}\int_{\theta=0}^{\theta=\pi/2}\int_{\rho=0}^{\rho=1}\rho\cos\theta\sin\varphi e^{\rho^2}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$
$=\displaystyle\left(\int_{\rho=0}^{\rho=1}\rho^3e^{\rho^2}\,\mathrm d\rho\right)\left(\int_{\theta=0}^{\theta=\pi/2}\cos\theta\,\mathrm d\theta\right)\left(\int_{\varphi=0}^{\varphi=\pi/2}\sin^2\varphi\,\mathrm d\varphi\right)=\frac\pi8$