   Chapter 15.5, Problem 24E

Chapter
Section
Textbook Problem

The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface. To determine

To find: The area of the given surface.

Explanation

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.

Given:

The surface is the cylinder y2+z2=1 intersects another cylinder x2+z2=1 .

Calculation:

Solve the given equations,

y2+z2=x2+z2y2=x2y=±x2y=±x

Since the given surface is the cylinder of radius 1, x ranges from 0 to 1 in the first quadrant of the xy-plane. So, the total area is 8 times the surface area of the above mentioned region in the first quadrant.

The partial derivatives fx and fy are,

fx=2x21x2=xzfy=0

Then, by the formula mentioned above, the area of the surface is given by,

A(S)=D(xz)2+(0)2+1dA=Dx2z2+0+1dA=Dx2+z2z2dA=01xx11x2dydx

Integrate it with respect to y and apply the limit

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