Triple Integrals in Spherical Coordinates

Question

Use spherical coordinates to to find the volume of a solid bounded above by
$x^2 + y^2 + z^2 = z$ and below by $z$ $=$ $\sqrt{x^2 + y^2}$

from the first equation I get p^2 = pcos(phi), and from the second one I get pcos(phi)= p|sin(phi)| , after that I'm completely lost as to find any bounds — thebottle394, Apr 24 '13 at 04:39

Mikasa · Answer 1 · 2013-04-24T08:04:11.697

3

Just hints for the limits:

$$V=4\int_{\phi=0}^{\pi/2}\int_{\theta=\pi/4}^{\pi/2}\int_{r=0}^{1/2} r^2\sin^2\theta dr d\theta d\phi$$ wherein $x^2+y^2+(z-1/2)^2=\frac{1}{4}$. See below:

enter image description here

edited Apr 24 '13 at 08:04

answered Apr 24 '13 at 05:28

Mikasa

67,374

The sphere is not centered at the origin. – Mhenni Benghorbal Apr 24 '13 at 05:53
@MhenniBenghorbal: Oh! :-0. I thought there is typo. Thanks. I'll edit it. Thanks. – Mikasa Apr 24 '13 at 05:54
@MhenniBenghorbal: Please have a look dear dr. I hope I could do that for the OP. – Mikasa Apr 24 '13 at 08:03
Beautiful graphic! – amWhy Apr 25 '13 at 01:54
@amWhy: It is like a Crown, Amy. Isn't it? – Mikasa Apr 25 '13 at 05:15
Yes it is, indeed! ;-) – amWhy Apr 25 '13 at 05:18
What software do you use to draw such graphics? – syntagma Jan 20 '15 at 20:46

Triple Integrals in Spherical Coordinates

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