# Find volume within sphere outside of Cylinder

1. Aug 2, 2009

### mknut389

1. The problem statement, all variables and given/known data
Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 16, above the xy plane, and outside the cone z = 3 $$\sqrt{{x^2+y^2}}$$.

2. Relevant equations
spherical system:
x=$$\rho$$cos$$\theta$$sin$$\phi$$
y=$$\rho$$sin$$\theta$$sin$$\phi$$
z=$$\rho$$cos$$\phi$$

cylindrical system
$$x^2+y^2=r^2$$
z=z

3. The attempt at a solution
I have tried this using both the spherical and cylindrical systems and arrived at the same answer, cylindrical is easiest here, so Ill use it to demonstrate what I have done

$$x^2+y^2+z^2=16$$
$$r^2+z^2=16$$
z=$$\sqrt{{16-r^2}}$$

z = 3 $$\sqrt{x^2+y^2}$$
z=3$$\sqrt{r^2}$$
z=3r

when z=0
0=$$\sqrt{16+r^2}$$
r=4
0=3r
r=0

Therefore the bounds are
z: $$\left[3r,\sqrt{16-r^2}\right]$$
r:$$\left[0,4\right]$$
$$\theta$$: $$\left[0,2\pi\right]$$

Which gives me
$$\int_{0}^{2\pi}\int_{0}^{4}\int_{3r}^{\sqrt{16-r^2}}r dzdrd\theta$$

After going through the steps i get
$$-(256\pi)/3$$

First off it is negative, so that tells me I am completely off base here, but when doing the same problem with spherical coordinates, I get $$(256\pi)/3$$. Since this answer is wrong, I am at a loss of what I should be doing. Please help.

Last edited: Aug 2, 2009
2. Aug 2, 2009

### mknut389

I solved it:

First the bounds had to be reworked for r in terms of Z
r: $$\left[z/3,\sqrt{16-z^2}\right]$$

Then by solving the two equations in terms of Z and setting them equal to each other, I determined the maximum value for r
$$\sqrt{16-r^2}=3r$$
$$r=(2*\sqrt{10})/5$$
after plugging that in and solving for z, z's maximum is 3.83863112788

Then using that as the bounds for Z and using the same bounds for $$\theta$$
The answer came out to be 127.137105725

Thanks to anyone that attempted to figure it out for me...