How Do You Correctly Set Up a Triple Integral Over a Sphere?

[PsychonautQQ]

Homework Statement

Triple Integral: x^2+y^2+z^2dV over the ball x^2+y^2+z^2 ≤ 9

Homework Equations

The Attempt at a Solution

so With my integral I had
Triple Integral: p^3sin∅dpd∅dθ
0≥p≥3
0≥∅≥∏
0≥θ≤2∏

Does this look like the correct integral? I swear it is! Yet my answer is wrong. I rebuke these foul math gods!

 

[pasmith]

Homework Statement

Triple Integral: x^2+y^2+z^2dV over the ball x^2+y^2+z^2 ≤ 9

Homework Equations

The Attempt at a Solution

so With my integral I had
Triple Integral: p^3sin∅dpd∅dθ
0≥p≥3
0≥∅≥∏
0≥θ≤2∏

Does this look like the correct integral? I swear it is! Yet my answer is wrong. I rebuke these foul math gods!

$$\iiint_{r \leq 3} r^2\,dV = \int_0^{2\pi} \int_0^{\pi} \int_0^3 (r^2) (r^2 \sin \theta) \,dr\,d\theta\,d\phi$$

 

[haruspex]

$$\iiint_{r \leq 3} r^2\,dV = \int_0^{2\pi} \int_0^{\pi} \int_0^3 (r^2) (r^2 \sin \theta) \,dr\,d\theta\,d\phi$$

That looks to me the same as PsychonautQQ posted, just with some of the ≤/≥ turned around the right way and with theta and phi swapped.
PsychonautQQ, what answer do you get and what is it supposed to be? Pls post your working in solving the integral.

 

[pasmith]

That looks to me the same as PsychonautQQ posted, just with some of the ≤/≥ turned around the right way and with theta and phi swapped.

And the correct expression for the volume element ...

 

[haruspex]

And the correct expression for the volume element ...

Ah yes - the extra r factor.