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...