Explanation Given: The provided integral is ∫ − ∞ ∞ e − x 2 d x . Formula used: Integration of a function x n is calculated as, ∫ x n = x n + 1 n + 1 + C By using Shell method, the volume of a solid revolving about y-axis is calculated. V = ∫ a b 2 π f ( x ) d x Calculation: The graph of the function y = e − x 2 is shown below, Using shell method to calculate the volume of solid, V = ∫ 0 ∞ 2 π x e − x 2 d x Substitute x 2 = t and differentiate it with respect to x as, 2 x d x = d t Using this result, solve the above integral as, V = π ∫ 0 ∞ 2 x e − x 2 d x = π ∫ 0 ∞ e − t d t = π [ e − t − 1 ] 0 ∞ = π [ − e − ∞ − ( − e − 0 ) ] Simplify the above equation further, to get; π [ − e − ∞ − ( − e − 0 ) ] = π [ 0 + 1 ] = π This implies that the volume of the solid is π cubic units

11th Edition

ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14, Problem 12PS

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To calculate: The volume of the solid generated by revolving the region in the first quadrant bounded by y=e−x2 about the y-axis. Using this result, also calculate the value of integral ∫−∞∞e−x2dx.

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Chapter 14, Problem 11PS

Chapter 14, Problem 14PS

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Chapter 14 Solutions

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The volume of the solid generated by revolving the region in the first quadrant bounded by y = e − x 2 about the y -axis. Using this result, also calculate the value of integral ∫ − ∞ ∞ e − x 2 d x .

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To calculate: The volume of the solid generated by revolving the region in the first quadrant bounded by y=e−x2 about the y-axis. Using this result, also calculate the value of integral ∫−∞∞e−x2dx.

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Chapter 14 Solutions