Young's Integral

Let $f(x)$ be a Real continuous monotonic strictly increasing function on the interval $[0,a]$ with $f(0)=0$ and $b\leq f(a)$, then

$$ab\leq\int_0^a f(x)\,dx+\int_0^b f^{-1}(y)\,dy,$$

where $f^{-1}(y)$ is the Inverse Function. Equality holds Iff $b=f(a)$.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1099, 1979.