How can I take integral just knowing the functiong graph?

Question

$f:[-2,4]\to[-1,2]$ is a bijective function.

How can I calculate $$\int_{-2}^{3}f(x)dx + \int_{-1}^{2}f^{-1}(x)dx$$ using the graph?

Answer 1

For $y=f(x)$, let $x=f^{-1}(y)$, then we want to calculate $$\int_{-2}^3 f(x)dx+\int_{f(-2)}^{f(3)} f^{-1}(y)dy$$ Suppose $x_0$ is where $f(x_0)=0$. Now $$\int_{-2}^3 f(x)dx+\int_{f(-2)}^{f(3)} f^{-1}(y)dy\\=\int_{-2}^{x_0}f(x)dx+\int_{x_0}^0f(x)dx+\int_0^3f(x)dx+\int_{-1}^{f(x_0)=0}f^{-1}(y)dy+\int_0^{f(0)}f^{-1}(y)dy+\int_{f(0)}^2f^{-1}(y)dy$$ Now we see that for example $$\int_0^3f(x)dx+\int_{f(0)}^2f^{-1}(y)dy=\text{area of rectangle }[0,3]\times[0,2]=6$$ or $$\int_{x_0}^0f(x)dx+\int_0^{f(0)}f^{-1}(y)dy=\text{area of rectangle }[x_0,0]\times[0,f(0)]$$ and so on.