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Solved: Volumes with infinite integrands Find the volume

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 38E Chapter 7.7

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 38E

Volumes with infinite integrands Find the volume of the described solid of revolution or state that it does not exist.

The region bounded by f(x) = (x2 − 1)−1/4 and the x-axis on the interval (1, 2] is revolved about the y-axis.

Step-by-Step Solution:
Step 1 of 3

Problem 38E

Volumes with infinite integrands Find the volume of the described solid of revolution or state that it does not exist.

The region bounded by f(x) = (x2 − 1)−1/4 and the x-axis on the interval (1, 2] is revolved about the y-axis.

Answer;

 

Step-1;

             The shell method ; If R is the region under the curve y = f(x)  on the interval [a , b] , then the volume of the solid  obtained by revolving R  about the y-axis is

                                           V = 2x f(x) dx.

Step-2

              Now , we have to find  out  the volume  of the  described  solid  under the region  bounded by  f(x) =  and the  x-axis  on the interval (1 , 2]  is revolved about the y-axis.

                          Consider , y = f(x) =

       The  visual representation  of the interval  is given below;

                                     

                         Therefore ,  the volume  of the  described  solid  under the region  bounded by  f(x) =  and the  x-axis  on the interval (1 , 2]  is revolved about the y-axis  is ;

                Volume (V) =  2x f(x) dx , since by the above formula.

                                   = 2x   dx , since f(x) =  .

                                   =   dx , since  = ……..(1)

    To evaluate this integral , let us use  substitution method .

 Step-3 ;

                   Consider ,   , then   = t

                          If x = 1 ,  the lower limit of  t =   =   = 0.

                         If x = 2 , the upper limit of t =   =  =

                                    Differentiate both sides with respect to x  , then  ;

                                                     

                                                             2x = 4

Step 2 of 3

Chapter 7.7, Problem 38E is Solved
Step 3 of 3

Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Solved: Volumes with infinite integrands Find the volume

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