Log In Start studying!

Select your language

Suggested languages for you:
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

Mean Value Theorem for Integrals

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.

Mean Value Theorem for Integrals geometric explanation area under the curve equal to rectangle StudySmarterThe 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.

Mean Value Theorem for Integrals area of rectangle equal to area under the curve StudySmarter

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 .

Mean Value Theorem for Integrals area of rectangle equal to area under the curve StudySmarter

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

  • 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

Frequently Asked Questions about 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 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?

Show answer

Answer

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

Show question

Question

State the Mean Value Theorem for integrals in words.

Show answer

Answer

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

More about Mean Value Theorem for Integrals
60%

of the users don't pass the Mean Value Theorem for Integrals quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.

Get FREE ACCESS to all of our study material, tailor-made!

Over 10 million students from across the world are already learning smarter.

Get Started for Free
Illustration