   Chapter 16.6, Problem 52E

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.52. The part of the surface z = cos(x2 + y2) that lies inside the cylinder x2 + y2 = 1

To determine

To find: The area of the part of the surface z=cos(x2+y2) that lies inside the cylinder x2+y2=1 .

Explanation

Given data:

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

z=cos(x2+y2)

The required surface lies inside the cylinder x2+y2=1 .

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=cos(x2+y2)

Calculation of zx :

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

zx=x[cos(x2+y2)]=2xsin(x2+y2)

Calculation of zy :

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

zy=y[cos(x2+y2)]=2ysin(x2+y2)

Calculation of surface area of plane:

Substitute [2xsin(x2+y2)] for zx and [2ysin(x2+y2)] for zy in equation (1),

A(S)=D1+[2xsin(x2+y2)]2+[2ysin(x2+y2)]2dA

A(S)=D1+4(x2+y2)sin2(x2+y2)dA (2)

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

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