<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] Hint:
from $\rho^2\le 1$ you have the limits for $\rho$, that is : $0\le \rho \le 1$
so you can find the limits for $z$ that is: $0\le z \le \sqrt{9-\rho^2}$
Finally, from the condition that $x,y$ are positive, you have the limits $0\le \theta \le \frac{\pi}{2}$
Can you do from this?
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