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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 5.5.100a
Chapter 5, Problem 5.5.100a

Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍

Verified step by step guidance
1
Step 1: Recognize that the given function ƒ is even, meaning ƒ(𝓍) = ƒ(-𝓍). This property will be useful in simplifying the integral.
Step 2: Analyze the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍. Notice that the integrand contains the term 𝓍, which is an odd function (𝓍 = -𝓍 when reflected about the origin).
Step 3: Recall the property of definite integrals: If the integrand is an odd function and the limits of integration are symmetric about zero (e.g., from -a to a), then the integral evaluates to 0. In this case, 𝓍ƒ(𝓍²) is an odd function because 𝓍 is odd and ƒ(𝓍²) is even.
Step 4: Conclude that the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍 = 0 due to the symmetry of the integrand and the limits of integration.
Step 5: No further computation is needed because the integral evaluates to 0 based on the properties of odd and even functions.

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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 as a function f(x) that satisfies the condition f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means that the integral of an even function over a symmetric interval, such as [-a, a], can be simplified. For example, if f is even, then ∫₋ₐ⁺ₐ f(x) dx = 2∫₀⁺ₐ f(x) dx.
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Substitution in Integrals

Substitution is a technique used in integration to simplify the process of finding an integral. It involves changing the variable of integration to make the integral easier to evaluate. For instance, if we let u = g(x), then the integral ∫ f(g(x)) g'(x) dx can be transformed into ∫ f(u) du, which may be simpler to solve.
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Definite Integrals

A definite integral represents the signed area under a curve defined by a function f(x) between two limits, a and b. It is denoted as ∫ₐᵇ f(x) dx and can be computed using the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) - F(a). This concept is crucial for evaluating integrals over specific intervals.
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Related Practice
Textbook Question

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.


∫₀⁴ (4𝓍― 𝓍²) d𝓍

Textbook Question

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).                                                                                           

                                                                                                                                                                                     

 (a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .                                                                                                                               

                                                                                                                                                                               

 <IMAGE>                                                                                                                                                                                                           

                                                                                                                                                                                     

 ƒ(t) = 4t + 2 , a = 0

Textbook Question

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



Textbook Question

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(a) ∫₄⁰ 3𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the 𝓍-direction and 2b in the y-direction is (𝓍²/a²) + (y² /b²) = 1.

(a) Let d² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ ―a, a] to show that the average value of d² is (a² + 2b²) /3 .

Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

ƒ(𝓍) = sin 𝓍 ; a = 0 , b = π/2 , c = π