Integrating the inverse of a function.
Solution 1:
In your setup, if $f(0)=0$ and $f(b)=a$ then $$\int_0^a f^{-1}(x)\, d x + \int_0^b f(x)\, d x = ab$$ that is the two integrals combine to the area of a rectangle.
In your setup, if $f(0)=0$ and $f(b)=a$ then $$\int_0^a f^{-1}(x)\, d x + \int_0^b f(x)\, d x = ab$$ that is the two integrals combine to the area of a rectangle.