   Chapter 15.8, Problem 23E

Chapter
Section
Textbook Problem

Use spherical coordinates.23. Evaluate ∫∫∫E (x2 + y2) dV, where E lies between the spheres x2 + y2 + z2 = 4 and x2 + y2 + z2 = 9.

To determine

To evaluate: The given triple integral by using spherical coordinates.

Explanation

Given:

The function is f(x,y,z)=x2+y2 .

The region B lies between the spheres x2+y2+z2=4 and x2+y2+z2=9 .

Formula used:

If f is a spherical region E given by aρb,αθβ,cϕd , then, Ef(x,y,z)dV=αβabcdf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdϕdρdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z  then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

The spherical coordinates (ρ,θ,ϕ) corresponding to the rectangular coordinates (x,y,z) is,

ρ=x2+y2+z2ϕ=cos1(zρ)θ=cos1(xρsinϕ)

Calculation:

By the given conditions, it is observed that ρ varies from 2 to 3, θ varies from 0 to 2π and ϕ varies from 0 to π . Use the formula mentioned above to change the given problem into spherical coordinates. Then, by equation (1), the value of the given triple integral is,

Bf(x,y,z)dV=2302π0π(x2+y2)dzdydx=2302π0π[(ρsinϕcosθ)2+(ρsinϕsinθ)2](ρ2sinϕ)dϕdθdρ=2302π0π[ρ2sin2ϕcos2θ+ρ2sin2ϕsin2θ](ρ2sinϕ)dϕdθdρ=2302π0πρ4sin3ϕ[cos2θ+sin2θ]dϕdθdρ

=2302π0πρ4sin3ϕdϕdθdρ

Use (2) to integrate and apply the limit values

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

Solve the equations in Exercises 126. 6x(x2+1)2(x2+2)48x(x2+1)3(x2+2)3(x2+2)8=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the mean for the following set of scores: 2, 7, 9, 4, 5, 3, 0, 6

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

What condition on and is necessary for the standard beta pdf to be symmetric?

Probability and Statistics for Engineering and the Sciences

35. Testing How many ways can a 10-question true-false test be answered?

Mathematical Applications for the Management, Life, and Social Sciences

The implied domain of is: (1, ∞) (−∞, 1) x ≠ 1 (−1, 1)

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

In Exercises 1320, plot the point on a set of coordinate axes. (1.2,3.4)

Finite Mathematics for the Managerial, Life, and Social Sciences 