Mean Value Theorem for Integrals

Definition

The Mean Value Theorem for Integrals states that if $f$ is continuous on the closed interval $[a, b]$, then there exists at least one number $c$ in $[a, b]$ such that the value of the function at $c$ is exactly equal to the average value of the function over the interval. $f(c) = f_{ave} = \frac{1}{b - a} \int_a^b f(x) \, dx$

• How to read: “The function f evaluated at c is equal to the average value of f, which is one divided by the quantity b minus a, times the integral from a to b of f of x with respect to x.”
• Meaning: Some point $c$ on the interval attains exactly the average height; a rectangle of width $(b-a)$ and height $f(c)$ has the same area as the region under the curve.

Why It Matters

The mean value theorem for integrals provides the mathematical ‘average’ of a dynamic system; it allows us to simplify complex, fluctuating data into a single, representative value for the purpose of analysis and decision-making.

Core Concepts

• Existence Guarantee: Like the Mean Value Theorem for derivatives, this is an existence theorem; it tells us that such a point $c$ exists but doesn’t necessarily tell us how to find it.
• Geometric Interpretation: There exists a rectangle with width $(b - a)$ and height $f(c)$ whose area is exactly equal to the area under the curve $y = f(x)$ from $a$ to $b$.
• Requirement of Continuity: The theorem relies on the function being continuous. If the function has jumps or breaks, it may never actually attain its average value.

Connected Concepts

• Average Rate of Change and Secant Lines
• The Intermediate Value Theorem
• Equivalence
• Average Value Of Function
• Fundamental Theorem of calculus
• Smoothing Effect
• First Principles Thinking
• Symmetry