I'm with you thus far. Now deduce what phi(z) is inside psi(y,z), just as you found psi(y,z) inside f(x,y,z).
Have to dash, but will add a pic later.
... Hi there. Yes, just in case a picture helps. Your work so far in balloon sculpture...
... and why not determine psi similarly...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!