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Real Analysis: Geometric Interpretation in Terms of Areas

Note: * = infinite

Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula:

the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0.

How can I provide a geometric interpretation of this formula in terms of areas and then prove this formula.

Do I use the Identity Criterion?

Solution Summary

A functional relation is proven. The solution is comprehensive.