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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.4.39
Chapter 5, Problem 5.4.39

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.


Ζ’(𝓍) = 8 ― 2𝓍 on [0, 4]

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Step 1: Recall the formula for the average value of a function on an interval [a, b]. The average value is given by: 1(b-a)∫a→bΖ’(𝓍)d𝓍. Here, a = 0 and b = 4.
Step 2: Compute the definite integral of Ζ’(𝓍) = 8 - 2𝓍 over the interval [0, 4]. Set up the integral: ∫0→4(8-2𝓍)d𝓍. Evaluate this integral step by step.
Step 3: Divide the result of the integral by the length of the interval (b - a = 4 - 0 = 4) to find the average value of the function on [0, 4]. This gives the average value of Ζ’(𝓍).
Step 4: Set Ζ’(𝓍) equal to the average value found in Step 3. Solve the equation 8-2𝓍=average_value for 𝓍.
Step 5: Verify that the solution(s) for 𝓍 lie within the interval [0, 4]. These are the points at which the function equals its average value on the given interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that if a function is continuous on a closed interval [a, b], then there exists at least one point c in (a, b) such that the function's value at c equals the average value of the function over that interval. This average value is calculated as (1/(b-a)) * ∫[a to b] f(x) dx.
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Average Value of a Function

The average value of a function f(x) over the interval [a, b] is defined as (1/(b-a)) * ∫[a to b] f(x) dx. This concept allows us to determine a single representative value of the function across the interval, which can then be compared to the function's actual values at specific points within that interval.
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Average Value of a Function

Finding Points of Intersection

To find points where a function equals its average value, we set the function f(x) equal to the average value calculated from the previous concepts. This involves solving the equation f(x) = average value, which may require algebraic manipulation or numerical methods to identify the specific x-values where this equality holds.
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Related Practice
Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 βˆ«β‚€β΄ √(16― 𝓍² ) d𝓍

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


βˆ«Ο€/β‚„^Β³Ο€/⁴ (cotΒ² 𝓍 + 1) d𝓍

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Textbook Question

The composite function Ζ’(g(𝓍)) consists of an inner function g and an outer function Ζ’. If an integrand includes Ζ’(g(𝓍)), which function is often a likely choice for a new variable u?

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Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 βˆ« 𝓍/(βˆ›π“ + 4) d𝓍

Textbook Question

Evaluate βˆ«β‚ƒβΈ Ζ’ β€²(t) dt , where Ζ’ β€² is continuous on [3, 8], Ζ’(3) = 4, and Ζ’(8) = 20 .

Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  βˆ« 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)