Triple integral in spherical coordinates

[hitemup]

Homework Statement

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

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

Homework Equations

[/B]
$$x = Rsin\phi cos\theta$$
$$y = Rsin\phi sin\theta$$
$$z = Rcos\phi$$

The Attempt at a Solution

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

Are the bounds correct?

 

[Noctisdark]

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

 

[LCKurtz]

Homework Statement

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

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

Homework Equations

[/B]
$$x = Rsin\phi cos\theta$$
$$y = Rsin\phi sin\theta$$
$$z = Rcos\phi$$

The Attempt at a Solution

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

Are the bounds correct?

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.

 

[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.

 

[LCKurtz]

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.

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$$
\int_{\theta \text{ limits}}\int_{\phi \text{ limits}}\int_0^{\rho \text{ on the top in terms of } \phi} \text{Integrand }\rho^2\sin\phi~d\rho d\phi d\theta$$

 

[Noctisdark]

EDIT : Sorry, i thaught they were cylindrical