According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=$

Mean Value Theorem for Integrals Explained

The Mean Value Theorem is a key concept in calculus that connects the average value of a function over an interval to a specific point within that interval. When applied to integrals, it helps us understand the relationship between the total area under a curve and the function's value at a particular point.

Continuous Function and Theorem Statement

For a function $f(x)$ to satisfy the conditions of the Mean Value Theorem for Integrals, it must be continuous over a closed interval $[a,b]$. If this condition is met, the theorem guarantees that there exists at least one value, denoted as $\xi$ (pronounced "xi"), within the open interval $(a,b)$. This special value $\xi$ has the property that the definite integral of $f(x)$ from $a$ to $b$ is equal to the product of the function's value at $\xi$ and the length of the interval $(b-a)$.

In simpler terms, you can find a point $\xi$ in the interval such that the rectangle formed by the height $f(\xi )$ and the width $(b-a)$ has an area exactly equal to the area under the curve of $f(x)$ from $a$ to $b$.

Integral Formula Derivation

The mathematical statement of the Mean Value Theorem for Integrals is given by the following formula:

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=f(\xi )(b-a)$

Let's break down the components of this formula:

- $\underset{a}{\overset{b}{\int }}f\left(x\right)dx$: This represents the definite integral of the function $f(x)$ over the interval $[a,b]$. It calculates the net signed area between the graph of $f(x)$ and the x-axis from $x=a$ to $x=b$.
- $f(\xi )$: This is the value of the function $f$ at the specific point $\xi$. This value is often referred to as the "average value" of the function over the interval $[a,b]$.
- $(b-a)$: This represents the length or width of the interval $[a,b]$.

So, the theorem essentially states that the total area under the curve is equal to the average height of the function multiplied by the width of the interval.

Option Analysis and Correct Form

Now, let's compare the established formula from the Mean Value Theorem for Integrals with the given options to identify the correct one:Therefore, based on the definition and application of the Mean Value Theorem for Integrals, the integral $\underset{a}{\overset{b}{\int }}f\left(x\right)dx$ is correctly represented by $f(\xi )(b-a)$.