Triple integral w/ spherical subsitution

[dumbfoundead]

Homework Statement

f(x) is a differentiable function let
$F(t)= \int\int\int_{x^2+y^2+z^2\leq t^2} f(x^2+y^2+z^2) dx dy dz$

compute F$^{'}$(t)

Homework Equations

x=p sin $\phi$ cos$\theta$
y= p sin $\phi$ sin$\theta$
z= p cos $\phi$

spherical bounds 0<p<t 0<$\phi$<$\Pi$ 0<$\theta$ < 2$\Pi$

p^2 sin$\phi$ = jacobian determinant

3. The attempt at a solution

carried through the substitution $\int\int\int f(p^2) p^2 sin \phi dp d\phi d\theta$

dont know how to evaluate $\int$f(p^2) sin$\phi$ d$\phi$?

 

[Pengwuino]

Be careful about putting those repeated copies of the template in your questions.

The remaining integral is trivial. Your function is independent of $\phi$ so the $\phi$ integration is trivial; it's just the integral of $sin(\phi)$