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# Solved: Volumes with infinite integrands Find the volume

ISBN: 9780321570567 2

## Solution for problem 38E Chapter 7.7

Calculus: Early Transcendentals | 1st Edition

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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.

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

Step 3 of 3

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