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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 5.4.27
Chapter 5, Problem 5.4.27

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
Ζ’(𝓍) = 1/(𝓍² + 1) on [―1, 1]

Verified step by step guidance
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Step 1: Recall the formula for the average value of a function Ζ’(𝓍) on an interval [a, b]. The average value is given by: 1(b-a)∫fx(x)dx. Here, a = -1 and b = 1.
Step 2: Set up the integral for the average value. Substitute Ζ’(𝓍) = 1/(𝓍² + 1) into the formula: 12∫x-111x2+1dx. Note that the factor 12 comes from 1(b-a), where b - a = 2.
Step 3: Evaluate the integral. The integral of 1x2+1 is a standard result in calculus. It is arctan(x). Apply this result to the integral: 12[arctan(x)]x-11.
Step 4: Compute the definite integral by substituting the limits of integration. Substitute x = 1 and x = -1 into arctan(x): 12[arctan(1)-arctan(-1)]. Recall that arctan(1) and arctan(-1) are standard values.
Step 5: Draw the graph of Ζ’(𝓍) = 1/(𝓍² + 1) on the interval [―1, 1]. The function is symmetric about the y-axis and decreases as |𝓍| increases. Indicate the average value as a horizontal line on the graph, representing the computed average value.

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Key Concepts

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

Average Value of a Function

The average value of a continuous function over a closed interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This represents the mean value of the function's outputs over the specified interval, providing insight into the function's overall behavior.
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Average Value of a Function

Definite Integral

A definite integral computes the accumulation of a function's values over a specific interval [a, b]. It is represented as ∫[a to b] f(x) dx and can be interpreted as the area under the curve of the function between the limits a and b, which is essential for finding the average value.
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Definition of the Definite Integral

Graphing Functions

Graphing a function involves plotting its output values against input values on a coordinate plane. This visual representation helps in understanding the function's behavior, identifying key features such as intercepts, maxima, minima, and the average value, which can be marked on the graph for clarity.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for Ζ’(𝓍)) = x + 1 on [0, 4] with n = 50.

Textbook Question

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals

∫(eΒ³Λ£ ⁺¹ d𝓍

Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dy ∫¹⁰ᡧ³ √(𝓍⁢ + 1) d𝓍

Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 βˆ«Ο€/₁₆^Ο€/⁸ 8 cscΒ² 4𝓍 d𝓍

Textbook Question

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of Ζ’(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and π“= 6

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Textbook Question

Integrals with sinΒ² 𝓍 and cosΒ² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 βˆ«β‚€^Ο€/⁴ cosΒ² 8ΞΈ dΞΈ