Just in case a picture helps... you want to fill up

... where T = total income, a = average income, p = population, and...

... is the product rule - straight continuous lines differentiating downwards with respect to t.

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus: standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!