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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the area of the part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 + y2.

To determine

To find: The area of the part of the sphere x2+y2+z2=4z that lies inside the paraboloid z=x2+y2 .

Explanation

Given data:

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

x2+y2+z2=4z

The equation of the paraboloid is given as follows.

z=x2+y2

Formula used:

Write the expression to find the surface area of the plane with z=f(x,y) .

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

Write the equation of part of the sphere as follows.

x2+y2+z2=4z (2)

Rewrite the expression as follows.

x2+y2+(z24z+4)4=0x2+y2+(z2)2=4(z2)2=4x2y2z2=4x2y2

Rearrange the equation.

z=2+4x2y2 (3)

Calculation of zx :

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

zx=x(2+4x2y2)=0+124x2y2(02x0)=124x2y2(2x)=x4x2y2

Calculation of zy :

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

zy=y(2+4x2y2)=0+124x2y2(002y)=2y24x2y2=y4x2y2

Calculation of surface area of plane:

Substitute (x4x2y2) for zx and (y4x2y2) for zy in equation (1),

A(S)=D1+(x4x2y2)2+(x4x2y2)2dA=D1+x24x2y2+y24x2y2dA=D4x2y2+x2+y24x2y2dA

Simplify the equation.

A(S)=D44(x2+y2)dA (4)

As the required part of the sphere lies inside the paraboloid z=x2+y2 , substitute z for (x2+y2) in equation (2).

z+z2=4zz23z=0z(z3)=0z=0(or)3

Substitute 3 for z in the equation of paraboloid z=x2+y2 as follows

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Chapter 16 Solutions

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