Figure 6

Two kinds of integrals for a monotonically decreasing function.

An integral along x-axis is${\int }_{y=f\left(x\right)}^{\left[A,B\right]}ydx=\underset{n\to \infty }{lim}\sum _{k=1}^n{y}_k\Delta x_k$

and that along y-axis is${\int }_{x=f^{-1}\left(y\right)}^{\left[B,A\right]}xdy=-\underset{n\to \infty }{lim}\sum _{k=1}^n{x}_k\Delta y_k.$

It holds${\int }_{x=f^{-1}\left(y\right)}^{\left[B,A\right]}xdy-x_{\text{A}}\left(y_{\text{A}}-y_{\text{B}}\right)={\int }_{y=f\left(x\right)}^{\left[A,B\right]}ydx-y_{\text{B}}\left(x_{\text{B}}-x_{\text{A}}\right),$

where y_A = f(x_A) and y_B = f(x_B).