Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4
(a) ∫₀¹ (4𝓍―2𝓍³) d𝓍
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8. Definite Integrals
Introduction to Definite Integrals
1:52 minutesProblem 5.2.69d
Textbook Question
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.
Verified step by step guidance1
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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1mKey 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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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.
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