Volume Integral of f over Sphere: Find Solution

[kidsmoker]

Homework Statement

Find the volume integral of the function $$f=x^{2}+y^{2}+z^{2}$$ over the region inside a sphere of radius R, centered on the origin.

Homework Equations

Spherical polars $$x=rsin(\theta)cos(\phi), y=rsin(\theta)sin(\phi), z=rcos(\theta)$$

Jacobian in spherical polars = $$r^2sin(\theta)$$

The Attempt at a Solution

When i work through it I end up with the triple integral

$$V=\int^{R}_{0}dr\int^{\pi}_{-\pi}d\phi\int^{\pi}_{-\pi}d\theta (r^{2}sin^{2}\theta cos^{2}\phi+r^{2}sin^{2}\theta sin^{2}\phi + r^{2}cos^{2}\theta)r^2sin\theta$$

but I'm not too sure whether this is right. Mainly I'm not sure about the limits of integration.

Is this correct please?

Thanks.

 

[Dick]

If you want to integrate over all theta, that's only theta in [-pi/2,pi/2], isn't it? And you can simplify the integrand a LOT. x^2+y^2+z^2=r^2.

 

[kidsmoker]

Ah yeah I didn't bother simplifying the integrand but I can see I should have done cos it would have made it a lot easier to type lol. I thought about it some more and understand why the limits are as you said now. Thanks!

 

[HallsofIvy]

Is there are reason why you titled this "surface integral"?

 

[kidsmoker]

Because I was tired :D Oops lol