   Chapter 16.6, Problem 61E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

30. If the supply function for a commodity is , what is the producer's surplus at x = 20?

Mathematical Applications for the Management, Life, and Social Sciences

Write an inequality for each graph:

Elementary Technical Mathematics

Simplify. 1aba 1cossincos

Trigonometry (MindTap Course List)

π does not exist

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 