Textbook Question
Working with area functions Consider the function ƒ and the points a, b, and c.
(b) Graph ƒ and A.
ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4
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 5.4.3b
Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18
(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .
Verified step by step guidance1
Step 1: Recall the definition of an even function. An even function satisfies ƒ(𝓍) = ƒ(-𝓍) for all 𝓍 in its domain. This symmetry will be important for solving the integral.
Step 2: Consider the integral ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍. Notice that the integrand 𝓍ƒ(𝓍) involves the product of 𝓍 and ƒ(𝓍).
Step 3: Analyze the symmetry of the integrand. Since ƒ(𝓍) is even, ƒ(𝓍) = ƒ(-𝓍). However, 𝓍 is an odd function because 𝓍 = -𝓍 when reflected about the origin. The product of an even function and an odd function is an odd function.
Step 4: Recall a key property of definite integrals: the integral of an odd function over a symmetric interval [−a, a] is always 0. This is because the positive and negative contributions cancel each other out.
Step 5: Conclude that ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 = 0 based on the symmetry of the integrand and the properties of odd functions over symmetric intervals.
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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 implies that the area under the curve from -a to 0 is equal to the area from 0 to a. This property is crucial for evaluating integrals involving even functions, as it simplifies calculations and allows for certain symmetries to be exploited.
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Exponential Functions
Properties of Definite Integrals
Definite integrals have several important properties, one of which is that the integral of an odd function over a symmetric interval around zero is zero. This is because the areas above and below the x-axis cancel each other out. Understanding these properties helps in evaluating integrals more efficiently, especially when dealing with functions that exhibit symmetry.
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Integration by Parts
Integration by parts is a technique used to integrate products of functions and is based on the product rule for differentiation. It is expressed as ∫u dv = uv - ∫v du, where u and v are differentiable functions. This method is particularly useful when integrating products of polynomials and other functions, allowing for the transformation of complex integrals into simpler forms.
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Related Practice
Textbook Question
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
b) Calculate g'(𝓍)
g(𝓍) = ∫₀ˣ sin² t dt
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Textbook Question
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
(b) Verify that A'(𝓍) = ƒ(𝓍).
ƒ(t) = 3t + 1 , a = 2
Textbook Question
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
ƒ(𝓍) = x² ― 1 on [2,5] ; n = 75
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].
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Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀⁴ (4𝓍― 𝓍²) d𝓍
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