Chapter 16.6, Problem 60E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Show that the parametric equations *x* = *a* cosh *u* cos *v*,* y* = *b* cosh *u* sin *v*,* z* = *c* sinh *u*, represent a hyperboloid of one sheet.

(b) Use the parametric equations in part (a) to graph the hyperboloid for the case *a* = 1, *b* = 2, *c* = 3.

(c) Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part (b) that lies between the planes *z* = −3 and *z* = 3.

(a)

To determine

**To show:** The parametric equations

Explanation

**Given data:**

The parametric equations are given as follows.

**Formula used:**

Write the expression for equation of hyperboloid of one sheet.

Here,

Write the parametric equations as follows.

Rewrite the parametric equations as follows.

Take square and add the expressions as follows.

Simplify the expression as follows

**(b)**

To determine

**To draw:** The graph of a hyperboloid of one sheet with the parametric equations

**(c)**

To determine

**To find:** The expression for surface area of the hyperboloid of one sheet with the parametric equations

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