Let $0<a<b $ and $ f: [a,b] \rightarrow [c,d]$ be an increasing bijection. Prove the following equality by area
$\int_{a}^{b}f(x)dx=bd-ac-\int_{c}^{d}f^{-1}(y)dy$.
Use integration by parts to prove the equality.
I honestly have no idea with this problem. I need full and detailed answer.