Spherical Coordinates Triple Integral

[qamptr]

I thought this question was elementary... but I apparently know less than I thought I did.

Homework Statement

Use spherical coordinates to evaluate $$\iiint_{E} x^{2}+y^{2}+z^{2}dV$$
Where E is the ball $$x^{2}+y^{2}+z^{2}\leq 16$$

Homework Equations

$$x^{2}+y^{2}+z^{2}=\rho^{2}$$

The Attempt at a Solution

$$\int^{2\pi}_{0}\int^{\pi}_{0}\int^{4}_{0}\left(\rho^{2}\right)\rho Sin \left( \phi \right) d\rho d\phi d\theta = 256\pi$$

which is apparently incorrect. Where am I going wrong?

 

[Hootenanny]

I thought this question was elementary... but I apparently know less than I thought I did.

Homework Statement

Use spherical coordinates to evaluate $$\iiint_{E} x^{2}+y^{2}+z^{2}dV$$
Where E is the ball $$x^{2}+y^{2}+z^{2}\leq 16$$

Homework Equations

$$x^{2}+y^{2}+z^{2}=\rho^{2}$$

The Attempt at a Solution

$$\int^{2\pi}_{0}\int^{\pi}_{0}\int^{4}_{0}\left(\rho^{2}\right)\rho Sin \left( \phi \right) d\rho d\phi d\theta = 256\pi$$

which is apparently incorrect. Where am I going wrong?

You may want to check your volume element .

 

[qamptr]

You may want to check your volume element .

Am I blind? I don't understand.

 

[qamptr]

You may want to check your volume element .

$$\left(\rho^{3}\right) \rho$$ instead of $$\left(\rho^{2} \right) \rho$$ gets me the right answer... but why?

 

[qamptr]

$$\left(\rho^{3}\right) \rho$$ instead of $$\left(\rho^{2} \right) \rho$$ gets me the right answer... but why?

Oh...