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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.2.E.71
Chapter 6, Problem 6.2.E.71

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.


Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br

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Step 1: Understand the problem. You are tasked with finding the area of the region bounded by two functions, f(x) = x^(1/n) and g(x) = x^n, on the interval [0,1]. Then, you need to evaluate the limit of this area as n approaches infinity and interpret the result.
Step 2: Set up the integral for the area. The area Aₙ is given by the integral of the difference between the upper function f(x) = x^(1/n) and the lower function g(x) = x^n over the interval [0,1]. The formula for the area is: An=∫01(x1/n-xn)dx
Step 3: Break the integral into two parts. The integral can be split into two separate integrals: An=∫01x1/ndx-∫01xndx. Evaluate each integral separately.
Step 4: Evaluate the first integral. For the integral ∫01x1/ndx, use the power rule for integration: ∫01xkdx=1k+1, where k = 1/n.
Step 5: Evaluate the second integral. For the integral ∫01xndx, use the same power rule for integration, where k = n. After evaluating both integrals, subtract the second result from the first to find Aₙ. Finally, take the limit as n approaches infinity to determine lim n→∞ Aₙ and interpret the result.

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Key Concepts

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

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. In this context, the area Aₙ can be found by integrating the difference between the functions f(x) and g(x) over the interval [0, 1]. Understanding how to set up and evaluate definite integrals is crucial for solving the problem.
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Limits

Limits are fundamental in calculus, particularly when analyzing the behavior of functions as they approach a certain point or infinity. In this question, evaluating lim n→∞ Aₙ requires understanding how the area Aₙ changes as n increases without bound, which can reveal insights about the functions involved.
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Function Behavior

Analyzing the behavior of the functions f(x) = x^(1/n) and g(x) = x^n as n approaches infinity is essential. As n increases, f(x) approaches 1 for x in (0, 1) and g(x) approaches 0. Understanding these behaviors helps in determining the area between the curves and interpreting the limit of Aₙ.
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