# Triple integral in spherical coordinates

1. May 16, 2015

### hitemup

1. The problem statement, all variables and given/known data

Evaluate
$$\int \int \int _R (x^2+y^2+z^2)dV$$

where R is the cylinder
$0\leq x^2+y^2\leq a^2$,
$0\leq z\leq h$

2. Relevant equations

$$x = Rsin\phi cos\theta$$
$$y = Rsin\phi sin\theta$$
$$z = Rcos\phi$$

3. The attempt at a solution

$$2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta \int_{0}^{h/cos\phi}dR R^4sin\phi$$

Are the bounds correct?

Last edited: May 16, 2015
2. May 16, 2015

### Noctisdark

Do you have to use spherical co-ordinates, why not cylindrical ones ?

3. May 16, 2015

### LCKurtz

This is not set properly if you insist on using spherical coordinates. You would have to break it up into two integrals, one where $\rho$ goes from $0$ to the side of the cylinder and another where it goes from $0$ to the top. A much better choice is cylindrical coordinates. Try setting it up that way and check back with us.

4. May 16, 2015

### hitemup

Yes I realized at one time it would be easier if I used cylindrical coordinates, since the region itself is a cylinder. But I just wanted to know how to proceed with spherical coordinates.

5. May 17, 2015

### LCKurtz

OK, so give it a go. First set up$$\int_{\theta \text{ limits}}\int_{\phi \text{ limits}}\int_0^{\rho \text{ on the sides in terms of } \phi} \text{Integrand } \rho^2\sin\phi~d\rho d\phi d\theta$$then add$$\int_{\theta \text{ limits}}\int_{\phi \text{ limits}}\int_0^{\rho \text{ on the top in terms of } \phi} \text{Integrand }\rho^2\sin\phi~d\rho d\phi d\theta$$

6. May 17, 2015

### Noctisdark

EDIT : Sorry, i thaught they were cylindrical