Textbook Question
Evaluating integrals Evaluate the following integrals.
β«βΒΉ π β’ 2Λ£Β²βΊΒΉ dπ
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.RE.15a
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(a) β«βββ΄ Ζ(π) dπ
Verified step by step guidance1
Step 1: Understand the symmetry properties of even and odd functions. An even function satisfies Ζ(π) = Ζ(-π), meaning it is symmetric about the y-axis. An odd function satisfies g(π) = -g(-π), meaning it is symmetric about the origin.
Step 2: Recall the property of definite integrals for even functions. If Ζ(π) is even, then β«βββ Ζ(π) dπ = 2β«ββ Ζ(π) dπ. This property will be used to evaluate β«βββ΄ Ζ(π) dπ.
Step 3: Substitute the given value of β«ββ΄ Ζ(π) dπ = 10 into the formula for even functions. Using the property, β«βββ΄ Ζ(π) dπ = 2β«ββ΄ Ζ(π) dπ.
Step 4: Simplify the expression by multiplying the given value of β«ββ΄ Ζ(π) dπ by 2. This will give the result for β«βββ΄ Ζ(π) dπ.
Step 5: Conclude that the integral β«βββ΄ Ζ(π) dπ depends entirely on the symmetry property of the even function and the given value of β«ββ΄ Ζ(π) dπ.
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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(βx) = 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.
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Exponential Functions
Odd Functions
An odd function satisfies the condition g(βx) = βg(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.
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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.
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Related Practice
Textbook Question
Area functions and the Fundamental Theorem Consider the function
Ζ(t) = { t if β2 β€ t < 0
tΒ²/2 if 0 β€ t β€ 2
and its graph shown below. Let F(π) = β«ββΛ£ Ζ(t) dt and G(π) = β«ββΛ£ Ζ(t) dt.
(e) Evaluate F ''(β1) and F ''(1). Interpret these values.
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Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(e) β«ββΒ² 3πΖ(π)dπ
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) Consider the linear function Ζ(π) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.
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Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(a) Find the area function A (π) = β«βΛ£ Ζ(t) dt using the Fundamental Theorem.
Ζ(π) = sin π ; a = 0 , b = Ο/2 , c = Ο
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Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(c) β«βββ΄ (4Ζ(π) β 3g(π))dπ
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