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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 5.5.100b
Chapter 5, Problem 5.5.100b

Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍

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1
Step 1: Recognize that the integral involves a substitution. The given integral is ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍. To simplify, let u = 𝓍³. Then, compute the derivative of u with respect to 𝓍: du/d𝓍 = 3𝓍², or equivalently, du = 3𝓍² d𝓍.
Step 2: Rewrite the integral in terms of u. Substitute u = 𝓍³ and du = 3𝓍² d𝓍 into the integral. The integral becomes (1/3) ∫ f(u) du, where the factor of 1/3 comes from the substitution.
Step 3: Adjust the limits of integration. When 𝓍 = -2, u = (-2)³ = -8. When 𝓍 = 2, u = (2)³ = 8. Therefore, the new limits of integration are from u = -8 to u = 8.
Step 4: Use the property of even functions. Since ƒ is an even function, ƒ(u) = ƒ(-u). This allows us to simplify the integral over symmetric limits. Specifically, ∫₋₈⁸ ƒ(u) du = 2 ∫₀⁸ ƒ(u) du.
Step 5: Substitute the given value of ∫₀⁸ ƒ(𝓍) d𝓍 = 9 into the equation. The integral becomes (1/3) * 2 * 9. Simplify this expression to find the final result.

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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 ƒ(−x) = ƒ(x) for all x in its domain. This symmetry about the y-axis implies that the area under the curve from -a to a is twice the area from 0 to a. Understanding this property is crucial for evaluating integrals involving even functions, as it simplifies calculations and allows for the use of symmetry.
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Substitution in Integration

Substitution is a technique used in integration to simplify the process by changing the variable of integration. It involves selecting a new variable, often denoted as u, which is a function of x, and transforming the integral accordingly. This method is particularly useful when dealing with composite functions, as it can make the integral more manageable and easier to solve.
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Definite Integrals

A definite integral represents the signed area under a curve between two specified limits, a and b. It is denoted as ∫ₐᵇ ƒ(x) dx and provides a numerical value that reflects the accumulation of quantities, such as area or volume. Understanding the properties of definite integrals, including their evaluation and the Fundamental Theorem of Calculus, is essential for solving problems involving area calculations and function analysis.
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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 sin (πt² ) dt ( a Fresnel integral) 

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Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(b) ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

                                                                                                                                                                    

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

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Textbook Question

Generalizing the Mean Value Theorem for Integrals Suppose ƒ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) ∫ₐˣ ƒ(t) dt + (𝓍―a) ∫ₓᵇg(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that ∫ₐᶜ ƒ(t) dt = ƒ(c) (b ― c)                                                                                                              

                                                                                                                                                                                

(Source: The College Mathematics Journal, 33, 5, Nov 2002)

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Textbook Question

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

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

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