Textbook Question
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(b) β«ββ΄ Ζ(π) 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.15c
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
Verified step by step guidance1
Step 1: Recall the symmetry properties of even and odd functions. An even function satisfies Ζ(π) = Ζ(βπ), and its integral over a symmetric interval [βa, a] is twice the integral over [0, a]. An odd function satisfies g(π) = βg(βπ), and its integral over a symmetric interval [βa, a] is 0.
Step 2: Break the given integral β«βββ΄ (4Ζ(π) β 3g(π)) dπ into two separate integrals: β«βββ΄ 4Ζ(π) dπ and β«βββ΄ β3g(π) dπ. This uses the linearity property of integrals.
Step 3: For the first term, β«βββ΄ 4Ζ(π) dπ, note that Ζ(π) is an even function. Therefore, β«βββ΄ Ζ(π) dπ = 2β«ββ΄ Ζ(π) dπ. Multiply this result by 4 to account for the coefficient.
Step 4: For the second term, β«βββ΄ β3g(π) dπ, note that g(π) is an odd function. The integral of an odd function over a symmetric interval [βa, a] is 0. Therefore, this term evaluates to 0.
Step 5: Combine the results from Step 3 and Step 4. The final integral β«βββ΄ (4Ζ(π) β 3g(π)) dπ simplifies to the result obtained from the first term, as the second term is 0.
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3mKey Concepts
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Even and Odd Functions
An even function satisfies the property f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, an odd function satisfies g(-x) = -g(x), indicating symmetry about the origin. These properties are crucial for evaluating integrals over symmetric intervals, as they allow simplifications based on the behavior of the functions.
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Properties of Functions
Properties of Definite Integrals
Definite integrals have specific properties that can simplify calculations. For instance, the integral of an even function over a symmetric interval [-a, a] is twice the integral from 0 to a, while the integral of an odd function over the same interval is zero. These properties help in evaluating integrals without direct computation.
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Linear Combination of Integrals
The linearity of integrals allows us to combine integrals of functions through addition and scalar multiplication. Specifically, β«(af(x) + bg(x))dx = aβ«f(x)dx + bβ«g(x)dx, where a and b are constants. This property is essential for evaluating integrals involving multiple functions, as it enables the separation of terms for easier computation.
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Related Practice
Textbook Question
Find the intervals on which Ζ(π) = β«βΒΉ (tβ3) (tβ6)ΒΉΒΉ dt is increasing and the intervals on which it is decreasing.
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Textbook Question
Properties of integrals Suppose β«ββ΄ Ζ(π) dπ = 6 , β«ββ΄ g(π) dπ = 4 and β«ββ΄ Ζ(π) dπ = 2 . Evaluate the following integrals or state that there is not enough information.
ββ«βΒΉ 2Ζ(π) dπ
Textbook Question
The velocity in ft/s of an object moving along a line is given by v = Ζ(t) on the interval 0 β€ t β€ 6 (see figure), where t is measured in seconds.
(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)
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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
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.
(a) β«βββ΄ Ζ(π) dπ