TITLE: How should I solve this triple integral?
QUESTION [3 upvotes]: Hi everybody I have a triple integral I can't solve:

$$\iiint \sqrt {x^2+y^2+z^2} \,dx \,dy \,dz $$

Which the region is between $z=\sqrt {x^2+y^2}$ and $z=4$ .
The question says after using the spherical coordinates the answer is:

$${a\Large\int} _{0}^{\pi/4} \frac{(\sin \phi)d\phi}{\cos^4\phi} $$

So I've used the spherical coordinates $0<r<4$ and $ 0<\theta<2\pi$ and $0<\phi<\frac{\pi}{2}$. The things I don't get is first why is $\phi$ varying between 0 and $\frac{\pi}{4}$ instead of $\frac{\pi}{2}$? And secondly where did that $\cos^4\phi$ came from?
By the way the question asks for the value of $a$.

REPLY [4 votes]: The plane $z=4$ has equation $r=4\sec \phi.$ So your limits for $r$ are $0\leq r \leq 4\sec \phi.$
{"set_name": "stack_exchange", "score": 3, "question_id": 3142266}