Let $0 < a < b$. Let $f > 0$ be continuous and strictly increasing on $[a, b]$. Prove that $\int_a^b f + \int_{f(a)}^{f(b)} f^{-1} = bf(b) - af(a)$
By drawing the graph I can see and prove the theorem holds. Area of Outer rectangle - inner rectangle gives the sum of two integrands. How to prove theoretically?
