Skip to main content
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.RE.15a
Chapter 5, Problem 5.RE.15a

Symmetry properties Suppose βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 10 and βˆ«β‚€β΄ g(𝓍) d𝓍 = 20. Furthermore, suppose Ζ’ is an even function and g is an odd function. Evaluate the following integrals.


(a) βˆ«β‚‹β‚„β΄ Ζ’(𝓍) d𝓍

Verified step by step guidance
1
Step 1: Understand the symmetry properties of even and odd functions. An even function satisfies Ζ’(𝓍) = Ζ’(-𝓍), meaning it is symmetric about the y-axis. An odd function satisfies g(𝓍) = -g(-𝓍), meaning it is symmetric about the origin.
Step 2: Recall the property of definite integrals for even functions. If Ζ’(𝓍) is even, then βˆ«β‚‹β‚β‚ Ζ’(𝓍) d𝓍 = 2βˆ«β‚€β‚ Ζ’(𝓍) d𝓍. This property will be used to evaluate βˆ«β‚‹β‚„β΄ Ζ’(𝓍) d𝓍.
Step 3: Substitute the given value of βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 10 into the formula for even functions. Using the property, βˆ«β‚‹β‚„β΄ Ζ’(𝓍) d𝓍 = 2βˆ«β‚€β΄ Ζ’(𝓍) d𝓍.
Step 4: Simplify the expression by multiplying the given value of βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 by 2. This will give the result for βˆ«β‚‹β‚„β΄ Ζ’(𝓍) d𝓍.
Step 5: Conclude that the integral βˆ«β‚‹β‚„β΄ Ζ’(𝓍) d𝓍 depends entirely on the symmetry property of the even function and the given value of βˆ«β‚€β΄ Ζ’(𝓍) d𝓍.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Even Functions

An even function is defined by the property that Ζ’(βˆ’x) = Ζ’(x) for all x in its domain. This symmetry about the y-axis implies that the area under the curve from -a to 0 is equal to the area from 0 to a. Therefore, when integrating an even function over a symmetric interval, the integral can be simplified to twice the integral from 0 to a.
Recommended video:
6:13
Exponential Functions

Odd Functions

An odd function satisfies the condition g(βˆ’x) = βˆ’g(x) for all x in its domain. This property indicates that the function is symmetric about the origin, leading to the conclusion that the integral of an odd function over a symmetric interval around zero is zero. Thus, when evaluating the integral of an odd function from -a to a, the contributions from the negative and positive sides cancel each other out.
Recommended video:
06:21
Properties of Functions

Definite Integrals and Symmetry

Definite integrals represent the net area under a curve between two points. When evaluating integrals of even and odd functions over symmetric intervals, the properties of these functions allow for simplifications. For even functions, the integral from -a to a can be expressed as twice the integral from 0 to a, while for odd functions, the integral from -a to a equals zero, highlighting the importance of symmetry in calculus.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If Ζ’ is symmetric about the line 𝓍 = 2 , then βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 2 βˆ«β‚€Β² Ζ’(𝓍) d𝓍.

64
views
Textbook Question

Function defined by an integral Let Ζ’(𝓍) = βˆ«β‚€Λ£ (t ― 1)¹⁡ (t―2)⁹ dt .

(c) For what values of 𝓍 does Ζ’ have local minima? Local maxima?

57
views
Textbook Question

Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n:

βˆ«β‚€ΒΉ 𝓍ⁿd𝓍 + βˆ«β‚€ΒΉ ⁿ√(𝓍d𝓍) = 1

70
views
Textbook Question

Area of regions Compute the area of the region bounded by the graph of Ζ’ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

                                                                                                                                                                                    

 Ζ’(𝓍) = 2 sin 𝓍/4 on [0, 2Ο€]

103
views
Textbook Question

Symmetry properties Suppose βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 10 and βˆ«β‚€β΄ g(𝓍) d𝓍 = 20. Furthermore, suppose Ζ’ is an even function and g is an odd function. Evaluate the following integrals.


(e) βˆ«β‚‹β‚‚Β² 3𝓍ƒ(𝓍)d𝓍

57
views
Textbook Question

Symmetry properties Suppose βˆ«β‚€β΄ Ζ’(𝓍) d𝓍 = 10 and βˆ«β‚€β΄ g(𝓍) d𝓍 = 20. Furthermore, suppose Ζ’ is an even function and g is an odd function. Evaluate the following integrals.


(c) βˆ«β‚‹β‚„β΄ (4Ζ’(𝓍) ― 3g(𝓍))d𝓍

64
views