<s>[INST] Calculate the triple integral: $\iiint_W(x^2+y^2)z\ dx\,dy\,dz$ I am trying to solve the following triple integral:
$$
\iiint_W(x^2+y^2)z\,dx\,dy\,dz \\
W=\{(x,y,z) \in \mathbb{R}: x^2+y^2+z^2 \le 9; x^2+y^2\le 1; x \ge 0; y \ge 0; z \ge 0\}
$$
From which I know that there are two surfaces:
$$
x^2+y^2+z^3 = 9 \rightarrow \text{Sphere of radius 3} \\
x^2+y^2=1 \rightarrow \text{Cylinder with a bade radius of 1}
$$
Converting this to Cylindrical Coordinates:
$$
\rho^2+z^2 \le 9 \implies \rho^2 \le 9-z^2 \implies \rho \le \sqrt{9-z^2} \\
\rho^2\le 1
$$
Where do I go from here?
 [/INST] The integral is getting calculated in 1st octant therefore:
$$\iiint_W(x^2+y^2)z\,dx\,dy\,dz=\iiint_W R^5\sin^{3}\theta\cos\theta dR d\theta d \phi$$also we know that for $0<\theta<\theta_0$ where $\theta_0=\sin^{-1}\frac{1}{3}$ we have $0<R<3$ and for $\frac{\pi}{2}>\theta>\theta_0$  we have $0<R<\frac{1}{\sin\theta}$. So we can split the integral by two:$$I=\int_{0}^{\frac{\pi}{2}}\int_{\theta_0}^{\frac{\pi}{2}}\int_{0}^{\frac{1}{\sin\theta}} R^5\sin^{3}\theta\cos\theta dR d\theta d \phi+\int_{0}^{\frac{\pi}{2}}\int_{0}^{\theta_0}\int_{0}^{3} R^5\sin^{3}\theta\cos\theta dR d\theta d \phi$$$$=\frac{1}{6}\int_{0}^{\frac{\pi}{2}}\int_{\theta_0}^{\frac{\pi}{2}}\frac{1}{\sin^{3}\theta}\cos\theta d\theta d \phi+\frac{243}{2}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\theta_0}\sin^{3}\theta\cos\theta d\theta d \phi=\frac{3\pi}{16}+\frac{\pi}{3}=\frac{25\pi}{48}$$
</s>