   Chapter 16.6, Problem 50E

Chapter
Section
Textbook Problem

Find the area of the surface.50. The part of the sphere x2 + y2 + z2 = b2 that lies inside the cylinder x2 + y2 = a2, where 0 < a < b

To determine

To find: The area of the part of the sphere x2+y2+z2=b2 that lies inside the cylinder x2+y2=a2 .

Explanation

Given data:

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

x2+y2+z2=b2

The required sphere lies inside the cylinder x2+y2=a2 .

Here, the limit of radius of the cylinder a is 0ab .

Formula used:

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

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

As the cylinder encloses the sphere separate portions of the sphere in the upper and lower halves, the required area is the two times of the area determined with the formula in equation (1).

Write the expression to find required original area of the sphere as follows.

A=2A(S) (2)

Write the equation of part of the sphere as follows.

x2+y2+z2=b2

Rewrite the expression as follows.

z=b2x2y2 (3)

Calculation of zx :

Take partial derivative for equation (3) with respect to x.

zx=x(b2x2y2)=12b2x2y2(2x)=xb2x2y2

Calculation of zy :

Take partial derivative for equation (3) with respect to y.

zy=y(b2x2y2)=12b2x2y2(2y)=yb2x2y2

Calculation of surface area of plane:

Substitute (xb2x2y2) for zx and (yb2x2y2) for zy in equation (1),

A(S)=D1+(xb2x2y2)2+(yb2x2y2)2dA=D1+x2b2x2y2+y2b2x2y2dA=D1+x2+y2b2x2y2dA=Db2x2y2+x2+y2b2x2y2dA

Simplify the expression as follows.

A(S)=Db2b2x2y2dA

A(S)=D(bb2x2y2)dA (4)

Consider the parametric equations for the cylinder x2+y2=a2 as follows

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