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.
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 :
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
Step 2: Add the bounds of integration to your answer from Step 1. We’re looking for “c” on the interval [0, 1], so:
Step 3: Set the formula from Step 3 equal to your original function, replacing all x’s in the function with c’s.
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:
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:
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.
 Larson, R. & Edwards, B. (2016). Calculus. Cengage Learning.
Stephanie Glen. "Mean Value Theorem for Integrals: Definition, Example" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/mean-value-theorem-for-integrals/
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