In our discussions on derivatives, you learned about the Mean Value Theorem - an important theorem that claims that a function will take on its average value over an interval at least once. The Mean Value Theorem also has an application for integrals that is a consequence of the Mean Value Theorem and the Fundamental Theorem of Calculus.
Formula and meaning of the Mean Value Theorem for Integrals
The Mean Value Theorem for integrals states that if a function f is continuous on the closed interval [a, b], then there is a number c such that
Clearly, the left-hand side of the equation is the area under the curve of f on the interval (a, b). The right-hand side can be thought of as the area of a rectangle. So, the theorem states that the area under the curve is equal to the area of a rectangle with a width of the interval (b - a) and a height equal to the average value of the function f. Rearranging this equation to solve for f(c), the average value, we find
Let us visualize the Mean Value Theorem for integrals geometrically.
The area under the curve of a function f on the interval [a, b] is equal to a rectangle with a width of b - a and a height of the average value of f, f(c) - StudySmarter Original
Proof of the Mean Value Theorem for Integrals
Consider the definition of an antiderivative where
By the Fundamental Theorem of Calculus
and 
Because F is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), we can apply the Mean Value Theorem, which says there is a number c such that 
and
Using the outcomes of the Fundamental Theorem of Calculus
Examples of the Mean Value Theorem for Integrals
Example 1
For the function 
over the interval [1, 4], find the value c (the x-value where f(x) takes on its average value).
Step 1: Make sure f(x) is continuous over the closed interval
Since f(x) is a polynomial, we know it is continuous over the interval [1, 4].
Step 2: Evaluate the integral of f(x) over the given interval
Step 3: Apply the Mean Value Theorem for integrals to find the average value of f(x) over the interval
So, the average value that f(x) takes on is 14.5.
In Step 2, we found that the area under the curve is 
. To find the area of the rectangle, we multiply the width by the height.
Thus, the Mean Value Theorem for integrals holds.
Step 4: Find the x-value of f(c)
Since 
and we want to find c, we can set f(x) equal to 14.5.
To solve for x, we apply the quadratic formula.
Since 
is outside of the interval, 
.
Example 2
For the function 
, find the x-value where f(x) takes on the average value over the interval 
Step 1: Make sure f(x) is continuous over the open interval
The function sin(x) is continuous everywhere.
Step 2: Evaluate the integral of f(x) over the given interval
Use your knowledge of the unit circle to solve the trigonometric equations! Remember, 
is just a multiple of 
.
Step 3: Apply the Mean Value Theorem for integrals to find the average value of f(x) over the interval
So, the average value that f(x) takes on is 
.
In Step 2, we found that the area under the curve is 
units2. To find the area of the rectangle, we multiply the width by the height.
units2
Thus, the Mean Value Theorem for integrals holds.
Step 4: Find the x-value of f(c)
Since 
and we want to find c, we can set f(x) equal to 
.
Solving this equation graphically, we find that 
.
The Mean Value Theorem for Integrals Calculation
As a reminder
Mean Value Theorem for Integrals - Key takeaways
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