Calculus How To

Mean Value Theorem for Integrals: Definition, Example

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The mean value theorem for integrals states that somewhere under the curve of a function, there is a rectangle with an area equal to the whole area under a curve.
mvt integral

Usually, questions concerning the mean value theorem for integrals ask you to find a value for c. To answer that, you need the formal definition of the theorem.

Formal Definition of the Mean Value Theorem for Integrals

A continuous function f on a closed, bounded interval [a, b] has at least one number c in the interval (a, b) for which [1]:
theorem mvt integrals

In English, it’s saying that the definite integral from a to b is going to be equal, at some point, to a rectangle. The area of the rectangle is found with the equation f(c)*(a – b), which is the function value at c multiplied by the interval length.

Let’s look at an example of how you can use the formal definition to find a value for c.

Example: How to Find Point “c”

Example question: Given the function f(x) = x(1 – x), what value of c satisfies the MVT for integrals on [0, 1]?

Step 1: Find the indefinite integral
example step 1

Step 2: Add the bounds of integration to your answer from Step 1. We’re looking for “c” on the interval [0, 1], so:
example step 2

Step 3: Set the formula from Step 3 equal to your original function, replacing all x’s in the function with c’s.
example step 3

We’re doing this because we’re looking for one specific point in the function that’s equal to the definite integral on the given bounds.

Step 4: Solve the right side of the equation (Step 3) for the integral bounds:
example step 4

Step 5: Expand the left side of the equation (Step 3) and then set this equal to Step 4:
c – c2 = 1/6

Step 6: Solve for c:
solution to example mvt
For brevity, I skipped writing down all of the algebra steps of solving for c, but you can find the steps here on Symbolab.

Related article: MVT for derivatives.


[1] Larson, R. & Edwards, B. (2016). Calculus. Cengage Learning.

Stephanie Glen. "Mean Value Theorem for Integrals: Definition, Example" From Calculus for the rest of us!

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