Explanation Given Information: The provided function is, f ( x ) = x 2 e − x . Formula used: The area under the curve f ( x ) between x = a and x = b is given by, A = ∫ a b f ( x ) d x The simple power rule for the differentiation is, d d x ( x n ) = n x n − 1 The formula for integration by parts is given by: ∫ u d v = u v − ∫ v d u Calculation: Consider the provided function, f ( x ) = x 2 e − x . Now, use the formula of integration by parts, ∫ u d v = u v − ∫ v d u To obtain the integral as, ∫ 0 ∞ x 2 e − x d x = − x 2 e − x | 0 ∞ + ∫ 0 ∞ 2 x e − x d x Split the second integrand into two parts and let, u = 2 x And, d v = e − x d x Then, d u = 2 d x And, v = ∫ e − x d x = − e − x Use the formula of integration by parts again as, ∫ 0 ∞ x 2 e − x d x = − x 2 e − x | 0 ∞ + ∫ 0 ∞ 2 x e − x d x = lim b → ∞

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 13.7, Problem 36E

To determine

To calculate: The area below the graph of y=f(x) and above the x axis for function f(x)=x2e−x and find the interval for which f(x)>0 and graph the area for the integral of f(x).

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Chapter 13 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 13.1 - CHECKPOINT 1. For the interval [0, 2]. determine...Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

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The area below the graph of y = f ( x ) and above the x axis for function f ( x ) = x 2 e − x and find the interval for which f ( x ) > 0 and graph the area for the integral of f ( x ) .

Solution Summary: The author calculates the area below the graph of y = f(x) and above the x axis for the function.

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To calculate: The area below the graph of y=f(x) and above the x axis for function f(x)=x2e−x and find the interval for which f(x)>0 and graph the area for the integral of f(x).

Chapter 13 Solutions

The area below the graph of y = f ( x ) and above the x axis for function f ( x ) = x 2 e − x and find the interval for which f ( x ) &gt; 0 and graph the area for the integral of f ( x ) .

Solution Summary: The author calculates the area below the graph of y = f(x) and above the x axis for the function.

Concept explainers

Videos

To calculate: The area below the graph of y=f(x) and above the x axis for function f(x)=x2e−x and find the interval for which f(x)>0 and graph the area for the integral of f(x).

Chapter 13 Solutions

The area below the graph of y = f ( x ) and above the x axis for function f ( x ) = x 2 e − x and find the interval for which f ( x ) > 0 and graph the area for the integral of f ( x ) .