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).

And now,

So

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...

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!