   Chapter 16.6, Problem 59E

Chapter
Section
Textbook Problem

(a) Show that the parametric equations x = a sin u cos v, y = b sin u sin v, z = c cos u, 0 ⩽ u ⩽ π, 0 ⩽ v ⩽ 2π, represent an ellipsoid.(b) Use the parametric equations in part (a) to graph the(c) Set up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b).

(a)

To determine

To show: The parametric equations x=asinucosv,y=bsinusinv,z=ccosu,0uπ,0v2π represent an ellipsoid.

Explanation

Given data:

The parametric equations are given as follows.

x=asinucosv,y=bsinusinv,z=ccosu,0uπ,0v2π

Formula used:

Write the expression for equation of ellipsoid.

x2a2+y2b2+z2c2=1 (1)

Here,

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

Write the parametric equations as follows.

x=asinucosv,y=bsinusinv,z=ccosu,0uπ,0v2π

Rewrite the parametric equations as follows.

xa=sinucosv,yb=sinusinv,zc=cosu

Take square and add the expressions as follows.

(xa)2+(yb)2+(zc)2=(sinucosv)2+(sinusinv)2+(cosu)2

Simplify the expression as follows

(b)

To determine

To draw: The graph of an ellipsoid with the parametric equations x=asinucosv,y=bsinusinv,z=ccosu,0uπ,0v2π for a=1,b=2 , and c=3 .

(c)

To determine

To find: The expression for surface area of the ellipsoid with the parametric equations x=asinucosv,y=bsinusinv,z=ccosu,0uπ,0v2π for a=1,b=2 , and c=3 .

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