   Chapter 16.6, Problem 60E

Chapter
Section
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 x=acoshucosv,y=bcoshusinv,z=csinhu represent a hyperboloid of one sheet.

Explanation

Given data:

The parametric equations are given as follows.

x=acoshucosv,y=bcoshusinv,z=csinhu

Formula used:

Write the expression for equation of hyperboloid of one sheet.

x2a2+y2b2z2c2=1 (1)

Here,

a , b , and c are scalar parameters (constant real values).

Write the parametric equations as follows.

x=acoshucosv,y=bcoshusinv,z=csinhu

Rewrite the parametric equations as follows.

xa=coshucosv,yb=coshusinv,zc=sinhu

Take square and add the expressions as follows.

(xa)2+(yb)2(zc)2=(coshucosv)2+(coshusinv)2(sinhu)2

Simplify the expression as follows

(b)

To determine

To draw: The graph of a hyperboloid of one sheet with the parametric equations x=acoshucosv,y=bcoshusinv,z=csinhu for a=1,b=2 , and c=3 .

(c)

To determine

To find: The expression for surface area of the hyperboloid of one sheet with the parametric equations x=acoshucosv,y=bcoshusinv,z=csinhu for a=1,b=2 , and c=3 .

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