Definition of Average in the Integral Sense

The formula for the average value of a function ($f(x)$) over an interval $[a,b]$ is given by:

$$\overline{f}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

In this formula:

• $a$ and $b$ are the lower and upper bounds of the interval over which you want to find the average of $f(x)$.
• $f(x)$ is the function whose average you want to calculate.
• ${\int }_a^{b}f(x)dx$ represents the definite integral of $f(x)$ over the interval $[a,b]$.
• $\frac{1}{b-a}$ scales the integral result to find the average value over the interval.

This formula essentially calculates the mean value of the function $f(x)$ over the specified interval.