Upper and Lower Bounds of Integral

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $f$ have a maximum of $M$ and a minimum of $m$ on $\left[{a \,.\,.\, b}\right]$.

Let $\displaystyle \int_a^b f \left({x}\right) \ \mathrm dx$ be the definite integral of $f \left({x}\right)$ over $\left[{a \,.\,.\, b}\right]$.

Then:
 * $\displaystyle m \left({b - a}\right) \le \int_a^b f \left({x}\right)\ \mathrm dx \le M \left({b - a}\right)$

Proof
This follows directly from the definition of definite integral.


 * From the Continuity Property it follows that $m$ and $M$ both exist.


 * The closed interval $\left[{a \,.\,.\, b}\right]$ is a subdivision of itself.


 * By definition, the upper sum is $M \left({b - a}\right)$, and the lower sum is $m \left({b - a}\right)$.

The result follows.