Problem

(P) Evaluate $\iiint_\Omega ze^{x^2+y^2+3z^2}\,\mathrm{d}x\mathrm{d}y\mathrm{d}z$, where $\Omega$ is the cylinder defined by $\Omega=\{(x,y,z):x^2+y^2\le 1 \text{ and } 0\le z\le 1\}$. Answer: $\underline{\quad(17)\quad}$.