Explanation Given information: The solid bounded by the region y = x 2 + 1 , and y = 9 − x 2 about the line y = − 1 . Find the intersection points of the curves by equating the curves as shown below: f ( x ) = g ( x ) (1) Here, the top curve function is f ( x ) and the bottom curve function is g ( x ) . Substitute 9 − x 2 for f ( x ) and x 2 + 1 for g ( x ) in Equation (1). 9 − x 2 = x 2 + 1 2 x 2 + 8 = 0 2 x 2 = 8 x 2 = 4 x = ± 2 Procedure to sketch the region bounded by the two curves is explained below: Draw the graph for the function y = x 2 + 1 by substituting different values for x . Similarly in the same graph plot for the function y = 9 − x 2 by substituting different values for x . Shade the region lies between x = − 2 and x = 2 . The region enclosed by the curves y = x 2 + 1 and y = 9 − x 2 is shown in Figure (1). Refer to Figure (1). Consider that the solid is a washer which is obtained by rotating region, about the line y = − 1 . Consider that the inner radius r is ( x 2 + 1 ) − ( − 1 ) = x 2 + 2 Consider that the outer radius R is ( 9 − x 2 ) − ( − 1 ) = 10 − x 2

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5, Problem 10RE

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To calculate: The volume of the solid obtained by rotating region.

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