   Chapter 16.6, Problem 45E

Chapter
Section
Textbook Problem

Find the area of the surface.45. The part of the surface z = xy that lies within the cylinder x2 + y2 = 1

To determine

To find: The area of the part of the surface z=xy that lies within the cylinder x2+y2=1 .

Explanation

Given data:

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

z=xy

The required surface lies within 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=xy (2)

Calculation of zx :

Take partial derivative for equation (2) with respect to x.

zx=x(xy)=y

Calculation of zy :

Take partial derivative for equation (2) with respect to y.

zy=y(xy)=x

Calculation of surface area of plane:

Substitute y for zx and x for zy in equation (1),

A(S)=D1+(y)2+(x)2dA

A(S)=D1+y2+x2dA (3)

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

x=rcosθ,y=rsinθ,0r1,0θ2π

Substitute rcosθ for x , rsinθ for y , and apply the limits in equation (3),

A(S)=02π011+(rsinθ)2+(rcosθ)2rdrdθ=02π01r1+r2(cos2θ+sin2θ)drdθ=02π01r1+r2(1)drdθ {cos2θ+sin2θ=1}=02π01r1+r2drdθ

Simplify the expression as follows

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