   Chapter 16.6, Problem 54E

Chapter
Section
Textbook Problem

Find, to four decimal places, the area of the part of the surface z = (1 + x2)/(1 + y2) that lies above the square |x| + |y| ⩽ 1. Illustrate by graphing this part of the surface.

To determine

To find: The area of the part of the surface z=1+x21+y2 and draw the graph of the part of the surface z=1+x21+y2 .

Explanation

Given data:

The equation of part of the surface is given as follows.

z=1+x21+y2 (1)

The required part of the surface lies above the square |x|+|y|1 .

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 (2)

Calculation of zx :

Take partial differentiation for equation (1) with respect to x.

zx=x(1+x21+y2)=11+y2x(1+x2)=11+y2(0+2x)=2x1+y2

Calculation of zy :

Take partial differentiation for equation (1) with respect to y.

zy=y(1+x21+y2)=(1+x2)y(11+y2)=(1+x2)[1(1+y2)2y(1+y2)]=(1+x2)[1(1+y2)2(0+2y)]

zy=2y(1+x2)(1+y2)2

Calculation of surface area of plane:

Substitute 2x1+y2 for zx and 2y(1+x2)(1+y2)2 for zy in equation (1),

A(S)=D1+(2x1+y2)2+[2y(1+x2)(1+y2)2]2dA (3)

As the required surface lies above the square |x|+|y|1 , the limits of y are written as follows

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