   Chapter 15.5, Problem 6E

Chapter
Section
Textbook Problem

Find the area of the surface.6. The part of the cylinder x2 + z2 = 4 that lies above the square with vertices (0, 0), (1.0), (0, 1). and (1,1)

To determine

To find: The area of given surface.

Explanation

Given:

The part of the cylinder, z2+x2=4 .

The region D lies above the square whose vertices are (0,0),(0,1),(1,0),(1,1) .

Formula used:

The surface area with equation z=f(x,y),(x,y)D , where fx and fy are continuous, is A(S)=D[fx(x,y)]2+[fy(x,y)]2+1dA .

Here, D is the given region.

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (1)

Calculation:

Solve the given equations,

z2+x2=4z2=4x2z=4x2

The partial derivatives fx and fy are,

fx=2x2(4x2)12=x(4x2)12fy=0

Then, the area of surface is given by,

A(S)=D(x

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