You're integrating z in a triple integral very like the first example here, Pauls Online Notes : Calculus III - Triple Integrals in Spherical Coordinates.
So you'll have, int_0^(arcsin(3/5)) int_0^(2*pi) int_0^5 1/2 r^3 sin(2p) dr dt dp - Wolfram|Alpha
And, just in case a picture helps for working through the triple from the inside out...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!