Only losing track. You have avoided using an integral sign, which hasn't helped. On the other hand, so have I...

... where (key in spoiler) ...

Spoiler:

Spoiler:

I hope that does help.

Also, here it is with your sub...

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!