Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.RE.15e

Chapter 5, Problem 5.RE.15e

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

Verified step by step guidance

Step 1: Understand the symmetry properties of the functions. An even function satisfies ƒ(𝓍) = ƒ(-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: Analyze the integral ∫₋₂² 3𝓍ƒ(𝓍)d𝓍. Notice that the integrand 3𝓍ƒ(𝓍) is a product of 𝓍 (an odd function) and ƒ(𝓍) (an even function). The product of an odd function and an even function is an odd function.

Step 3: Recall a key property of definite integrals for odd functions: ∫₋𝓪^𝓪 h(𝓍)d𝓍 = 0 if h(𝓍) is an odd function. This property applies because the contributions from the interval [-𝓪, 0] and [0, 𝓪] cancel each other out.

Step 4: Conclude that the integrand 3𝓍ƒ(𝓍) is odd, and the integral ∫₋₂² 3𝓍ƒ(𝓍)d𝓍 evaluates to 0 based on the symmetry property of odd functions.

Step 5: Summarize the reasoning: The integral evaluates to 0 because the integrand is an odd function and the limits of integration are symmetric about the origin.

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Key Concepts

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

Even and Odd Functions

An even function is defined by the property f(-x) = f(x) for all x in its domain, which means its graph is symmetric about the y-axis. Conversely, an odd function satisfies g(-x) = -g(x), indicating that its graph is symmetric about the origin. Understanding these properties is crucial for evaluating integrals, as they can simplify calculations by exploiting symmetry.

Properties of Definite Integrals

Definite integrals have specific properties that can simplify their evaluation. For instance, the integral of an even function over a symmetric interval [-a, a] can be expressed as twice the integral from 0 to a. In contrast, the integral of an odd function over a symmetric interval is zero. These properties are essential for solving integrals involving even and odd functions.

Integration Techniques

Integration techniques involve various methods for calculating integrals, including substitution, integration by parts, and recognizing patterns in functions. In this context, recognizing the symmetry of the functions involved allows for the application of specific techniques that can simplify the evaluation of the integral, particularly when combined with the properties of even and odd functions.

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Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

Textbook Question

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 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

Sigma notation Evaluate the following expressions.

(a) 10

∑ κ

κ=1

Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

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Key Concepts

Even and Odd Functions

Properties of Definite Integrals

Integration Techniques

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍