# Volume using spherical coordinates

Hello. Here is the problem I am currently having difficulties with:
"find the volume of the solid that lies inside the cone z^2 = 3x^2 + 3y^2 and between spheres x^2 + y^2 + z^2 = 1 and x^2 + y^2 + z^2 = 9"

I know that this integral needs to be setup in spherical coordinates... Here is the integral I came up with. I'm not sure if it is correct though...
v = integral from 1 to 3 integral from 0 to 2pie integral from pie/4 to pie/2
p^2 sin(phi) dp d(phi) d(theta)
does this seem correct?

let me try to put it in LaTeX format... (sorry if it dosen't work..)

$$V=\int_1^3 \int_0^\Pi \int_\frac{\pi}{4}^\frac{\pi}{2} \rho^2 \sin\phi dpd\phi d\theta$$

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## Answers and Replies

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arildno
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Where do you get your angle values from??

When doing coordinate changes, it is always helpful to restate your equations in terms of your new coordinates.
First, the general transformation relations from Cartesian to polar:
$$x=r\sin\phi\cos\theta,y=r\sin\phi\sin\theta,z=r\cos\phi, 0\leq\theta\leq{2}\pi,0\leq\phi\leq\pi,0\leq{r}$$

Now, restatement of your equations delineating your region:
$$r^{2}=1, r^{2}=9,\tan^{2}\phi=\frac{1}{3}$$

What does this tell you?