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Mean Value Theorem for Integrals

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 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 .

As a reminder

Mean Value Theorem for Integrals - Key takeaways

• 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

• Geometrically speaking, the area under the curve is equal to the area of a rectangle with a width of b - a and a height of the average value of f(x), f(c)

• The Mean Value Theorem for integrals is a consequence of the Mean Value Theorem for derivatives and the Fundamental Theorem of Calculus

The Mean Value Theorem for integrals states that if a function f is continuous on the closed interval [a, b], then the area under the curve is equal to the are of a rectangle with width b - a and height equal to the average value of the function f.

To use the Mean Value Theorem, integrate the function over the given interval (a, b). Multiply the area under the curve by 1/(b-a) to find the average value over the given interval.

To find the value c, apply the Mean Value Theorem for integrals to find the function value at c. Then, set f(x) equal to f(c) and solve for x.

A simple example of the Mean Value Theorem for integrals is the function f(x)=x over the interval [0, 1] has an average value of 1/2 at x = 1/2. This means that the area under the of f(x) over the interval [0, 1] is equal to the area of a rectangle with a width of 1 and a height of 1/2.

The formula for the Mean Value Theorem for integrals says that the definite integral from a to b is equal to f(c)(b - a) for some c value.

Final Mean Value Theorem for Integrals Quiz

Question

The Mean Value Theorem for integrals is derived from which two theorems?

• The Mean Value Theorem for derivatives
• The Fundamental Theorem of Calculus

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Question

State the Mean Value Theorem for integrals in words.

The area under a function curve is equal to the area of a rectangle whose width is the interval of the function and whose height is the average value of the function over the interval.

Show question

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