   Chapter 16.6, Problem 53E

Chapter
Section
Textbook Problem

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.53. The part of the surface z = ln(x2 + y2 + 2) that lies above the disk x2 + y2 ⩽ 1

To determine

To find: The area of the part of the surface z=ln(x2+y2+2) that lies above the disk x2+y21 .

Explanation

Given data:

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

z=ln(x2+y2+2)

The required surface lies above the disk x2+y21 .

Formula used:

Write the expression to find the surface area of the plane.

A(S)=D1+(zx)2+(zy)2dA (1)

Write the equation of part of the surface as follows.

z=ln(x2+y2+2)

Calculation of zx :

Substitute ln(x2+y2+2) for z in the expression zx ,

zx=x[ln(x2+y2+2)]=1x2+y2+2(2x)=2xx2+y2+2

Calculation of zy :

Substitute ln(x2+y2+2) for z in the expression zy ,

zy=y[ln(x2+y2+2)]=1x2+y2+2(2y)=2yx2+y2+2

Calculation of surface area of plane:

Substitute 2xx2+y2+2 for zx and 2yx2+y2+2 for zy in equation (1),

A(S)=D1+(2xx2+y2+2)2+(2yx2+y2+2)2dA=D1+4x2(x2+y2+2)2+4y2(x2+y2+2)2dA

A(S)=D1+4(x2+y2)(x2+y2+2)2dA (2)

Consider the parametric equations for the cylinder x2+y2=1 as follows

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