   Chapter 16.6, Problem 51E

Chapter
Section
Textbook Problem

If the equation of a surface S is z = f(x, y), where x2 + y2 ⩽ R2, and you know that |fx| ⩽ 1 and |fy| ⩽ 1, what can you say about A(S)?

To determine

To find: The range of area of the surface with the equation z=f(x,y) .

Explanation

Given data:

The equation of surface is given as follows.

z=f(x,y)

The equation of circle with the parameters x and y is given as follows.

x2+y2R2

Here,

R is the radius of the circle.

The range of fx is given as follows.

|fx|1 (1)

The range of fy is given as follows.

|fy|1 (2)

Formula used:

Write the expression to find the surface area of the plane with the equation z=f(x,y) .

A(S)=D1+(zx)2+(zy)2dA (3)

Write the expression to find the area of circle with the radius R .

A=πR2 (4)

Calculation of zx :

Substitute f(x,y) for z in the expression zx .

zx=x[f(x,y)]=fx

Calculation of zy :

Substitute f(x,y) for z in the expression zy .

zy=y[f(x,y)]=fy

Calculation of surface area of plane:

Substitute fx for zx and fy for zy in equation (3),

A(S)=D1+(fx)2+(fy)2dA (5)

Rewrite the expression in equation (1) as follows.

0fx1

Take square and rewrite the expression as follows.

(0)2(fx)2(1)2

0(fx)21 (6)

Rewrite the expression in equation (2) as follows.

0fy1

Take square and rewrite the expression as follows.

(0)2(fy)2(1)2

0(fy)21 (7)

Add equations (6) and (7) as follows

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