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Volume of a hyperbolic cap Consider the region R bounded

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

Solution for problem 81E Chapter 10.4

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 81E

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x2/a2 − y2/b2 = 1 and the vertical line through the right focus.

a. What is the volume of the solid that is generated when R is revolved about the x-axis?

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

Step-by-Step Solution:

Solution 81EStep 1 :

Given : .a. To calculate the volume of the hyperbolic cap, we know that the volume of the solid generated by revolving the curve y=f(x)  about the x-axis is given by

Here the volume generated by is calculated by putting and using the limit from vertex (a,0) to focus (ae,0) as follows                    

Hence the volume of the solid that is generated when R is revolved about the x-axis is :  .

Step 2 of 2

Chapter 10.4, Problem 81E is Solved
Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Volume of a hyperbolic cap Consider the region R bounded

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