Suppose $f'$ is continuous and positive on $[a, b]$. Prove $\int_a^b f(x) dx + \int_{f(a)}^{f(b)} f^{-1} (y) dy = b f(b) - a f(a)$
Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$