Explanation Calculation: The functions of the solid are as follows. x = 3 y 2 − 2 (1) x = y 2 (2) Rearrange the Equation (1). 3 y 2 = x + 2 y 2 = x + 2 3 y = x + 2 3 (3) Rearrange the Equation (2). y = x (4) The total volume of the solid ( V ) is as shown below. V = V 1 + V 2 (5) Here, V 1 is the volume of the solid 1 and V 2 is the volume of the solid 2. Disk method: On the interval [ − 2 , 0 ] , the radius of the disk is as follows. R 1 ( x ) = x + 2 3 Here, R 1 ( x ) is the radius of the solid 1. On the interval [ 0 , 1 ] , the radius of the washer swept out by the line segment are as follows. R 2 ( x ) = x + 2 3 r 2 ( x ) = x Here, R 2 ( x ) is the outer radius of the solid 2 and r 2 ( x ) is the inner radius of the solid 2. Calculate the total volume of the solid ( V ) using the disk method: V = V 1 + V 2 = ∫ − 2 0 π { [ R 1 ( x ) ] 2 } d x + ∫ 0 1 π { [ R 2 ( x ) ] 2 − [ r 2 ( x ) ] 2 } d x Substitute x + 2 3 for R 1 ( x ) , x + 2 3 for R 2 ( x ) , and x for r 2 ( x ) . V = ∫ − 2 0 π { [ x + 2 3 ] 2 } d x + ∫ 0 1 π { [ x + 2 3 ] 2 − [ x ] 2 } d x = ∫ − 2 0 π ( x + 2 3 ) d x + ∫ 0 1 π ( x + 2 3 − x ) d x = ∫ − 2 0 π ( x + 2 3 ) d x + ∫ 0 1 π ( − 2 x + 2 3 ) d x Hence, the integrals would be required in disk method is 2. Washer method: On the interval [ − 2 , 0 ] , the radius of the solid are as follows. R 1 ( x ) = x + 2 3 r 1 ( x ) = 0 Here, R 1 ( x ) is the outer radius of the solid 1 and r 1 ( x ) is the inner radius of the solid 1. On the interval [ 0 , 1 ] , the radius of the solid are as follows. R 2 ( x ) = x + 2 3 r 2 ( x ) = x Here, R 2 ( x ) is the outer radius of the solid 2 and r 2 ( x ) is the inner radius of the solid 2

University Calculus: Early Transcendentals Plus MyLab Math -- Access Card Package (3rd Edition) (Integrated Review Courses in MyMathLab and MyStatLab)

3rd Edition

ISBN: 9780321999573

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 6.2, Problem 39E

To determine

Calculate the volume of the solid.

Calculate the number of integrals would be required in each method (disk, washer, and shell).

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Chapter 6.2, Problem 38E

Chapter 6.2, Problem 40E

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Calculate the volume of the solid. Calculate the number of integrals would be required in each method (disk, washer, and shell).