Let $f$ be a strictly decreasing function. Then $\int_{a}^bf^{-1}=bf^{-1}(b)-af^{-1}(a)+\int_{f^{-1}(b)}^{f^{-1}(a)}f $ I'm trying to prove the fact that if $f$ is a strictly decreasing function, then:$$\int_{a}^bf^{-1}=bf^{-1}(b)-af^{-1}(a)+\int_{f^{-1}(b)}^{f^{-1}(a)}f $$
I have already proven it for strictly increasing functions. In that case, I made a sketch so I could understand the integral with geometry, and then, using the partitions: $$P=({t_0=a,t_1,...,t_n=b})$$
$$P'=(f^{-1} (t_0)=f^{-1}(a),f^{-1}(t_1),...,f^{-1}(t_n)=f^{-1}(b))$$ I computed $L(f^{-1},P)$+$U(f,P')$ and the rest of the proof was easy to develop.
However, I can't prove it when $f$ is strictly decreasing, neither can I see it geometrically.
Any advice will be appreciated.

The lighter region is the integral on your left-hand side. 

You have $$\int_a^bf^{-1}=+U-V+W$$
