(a) To determine Calculate the volume of the solid generated by revolving the region about the line y = 1 . Explanation Given information: The solid generated by revolving the region bounded by the parabola y = x 2 and the line y = 1 . Calculation: Provide the Equation of parabola as shown below. y = x 2 Sketch the region in the second quadrant bounded by the parabola y = x 2 and the line y = 1 as shown in Figure 1. Refer to Figure 1. The outer radius R ( x ) = 1 − x 2 . The inner radius r ( x ) = 0 . The region lies between − 1 to 1. Calculate the volume of the solid using the formula: V = ∫ a b π ( [ R ( x ) ] 2 − [ r ( x ) ] 2 ) d x (1) Substitute 1 − x 2 for R ( x ) , 0 for r ( x ) , − 1 for a , and 1 for b in Equation (1). V = ∫ − 1 1 π ( ( 1 − x 2 ) 2 − 0 2 ) d x = π ∫ − 1 1 ( 1 + x (b) To determine Calculate the volume of the solid generated by revolving the region about the line y = 2 . (c) To determine Calculate the volume of the solid generated by revolving the region about the line y = − 1 .

Thomas' Calculus - MyMathLab Integrated Review

14th Edition

ISBN: 9780134786223

Author: Hass

Publisher: PEARSON

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Question

Chapter 6.1, Problem 51E

(a)

To determine

Calculate the volume of the solid generated by revolving the region about the line y=1.

(b)

To determine

Calculate the volume of the solid generated by revolving the region about the line y=2.

(c)

To determine

Calculate the volume of the solid generated by revolving the region about the line y=−1.

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Chapter 6.1, Problem 50E

Chapter 6.1, Problem 52E

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