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 point $c$ in $(a, b)$ such that the integral of $f$ from $a$ to $b$ equals $f(c)$ times the length of the interval. Mathematically, this is expressed as $\int_a^b f(x) \, dx = f(c) (b - a)$.

5 Must Know Facts For Your Next Test

1. The theorem guarantees the existence of a point $c$ but does not provide a method to find it.
2. It applies only to functions that are continuous on the given closed interval.
3. The theorem is a direct consequence of the Fundamental Theorem of Calculus.
4. If $f(x)$ is constant, then any point in $[a, b]$ can serve as $c$.
5. The value of $f(c)$ represents the average value of the function over the interval.

Review Questions

• What conditions must be met for the Mean Value Theorem for Integrals to apply?
• How is the Mean Value Theorem for Integrals related to finding an average value?
• Can you use this theorem if a function has discontinuities in the interval?

Related terms

Fundamental Theorem of Calculus:

A principle that links differentiation and integration, stating that differentiation and integration are inverse processes.

Average Value of a Function: The average value of a function $f(x)$ over an interval $[a, b]$, calculated as $\frac{1}{b-a} \int_a^b f(x) \, dx$.

Continuous Function: A function without breaks, jumps, or interruptions in its domain.