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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 5.2.69d
Chapter 5, Problem 5.2.69d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍 = βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍, then Ζ’ is a constant function. 

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Step 1: Begin by understanding the integral notation. The expression βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍 represents the definite integral of the function Ζ’(𝓍) over the interval [a, b]. This computes the net area under the curve of Ζ’(𝓍) between x = a and x = b.
Step 2: Analyze the given statement. The statement claims that if βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍 = βˆ«β‚α΅‡ Ζ’(𝓍) d𝓍, then Ζ’ must be a constant function. This implies that the equality of the integrals is being used to infer something about the nature of the function Ζ’.
Step 3: Consider whether the equality of the integrals necessarily implies that Ζ’ is constant. Recall that the definite integral depends on the values of Ζ’(𝓍) over the interval [a, b], but it does not directly indicate whether Ζ’ is constant. For example, two different functions can have the same integral value over the same interval.
Step 4: Provide a counterexample to disprove the statement if necessary. For instance, consider two functions ƒ₁(𝓍) = 𝓍 and Ζ’β‚‚(𝓍) = 𝓍² - 𝓍 over a specific interval [a, b]. Compute their integrals and observe that they may yield the same result, even though neither function is constant.
Step 5: Conclude that the statement is false. The equality of the integrals does not necessarily imply that Ζ’ is a constant function. The integral only provides information about the net area under the curve, not the specific behavior of the function across the interval.

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The value of a definite integral can provide insights into the behavior of the function, such as total accumulation or net change over the interval.
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Definition of the Definite Integral

Constant Function

A constant function is a function that always returns the same value regardless of the input variable. Mathematically, it can be expressed as f(x) = c, where c is a constant. In the context of integrals, if the integral of a function over an interval is equal to itself, it does not necessarily imply that the function is constant; rather, it indicates that the area under the curve remains unchanged.
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Counterexample

A counterexample is a specific case that disproves a general statement or proposition. In calculus, providing a counterexample can effectively demonstrate that a certain condition does not hold true for all functions. For instance, if two integrals are equal, one can find a non-constant function that satisfies this equality, thus serving as a counterexample to the claim that the function must be constant.
Related Practice
Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


βˆ«β‚ƒβΆ (1―2𝓍) d𝓍 ; n = 6

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


{Use of Tech} Ζ’(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

Textbook Question

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) βˆ«β‚€^Ο€/¹⁢ sec Β² 4𝓍 d𝓍

Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

Ζ’(𝓍) = xΒ² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums. 

Textbook Question

Area functions The graph of Ζ’ is shown in the figure. Let A(x) = βˆ«β‚€Λ£ Ζ’(t) dt and F(x) = βˆ«β‚‚Λ£ Ζ’(t) dt be two area functions for Ζ’. Evaluate the following area functions.

(d) F(8)

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Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) βˆ«β‚€Β³ 5Ζ’(𝓍) d𝓍