Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.2.69d

Capitolo 5, Problema 5.2.69d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function.

Guida verificata passo dopo passo

Step 1: Begin by understanding the integral notation. The expression ∫ₐᵇ ƒ(𝓍) d𝓍 represents the definite integral of the function ƒ(𝓍) over the interval [a, b]. This computes the net area under the curve of ƒ(𝓍) between x = a and x = b.

Step 2: Analyze the given statement. The statement claims that if ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ 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 ƒ.

Step 3: Consider whether the equality of the integrals necessarily implies that ƒ is constant. Recall that the definite integral depends on the values of ƒ(𝓍) over the interval [a, b], but it does not directly indicate whether ƒ 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 ƒ₂(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 ƒ 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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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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Percorso guidato

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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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Exponential Functions

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.

Pratica correlata

Domanda del libro di testo

{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.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

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Domanda del libro di testo

{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.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

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Domanda del libro di testo

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

(d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then

B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

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Domanda del libro di testo

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(8)

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Domanda del libro di testo

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(d) 1 + 1/2 + 1/3 + 1/4

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Domanda del libro di testo

Sigma notation Evaluate the following expressions.

(d) 5

∑ (1 + n²)

n=1

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