   Chapter 15.8, Problem 30E

Chapter
Section
Textbook Problem

Use spherical coordinates.30. Find the volume of the solid that lies within the sphere x2 + y2 + z2 = 4, above the xy-plane, and below the cone z = x 2 + y 2 .

To determine

To find: The volume of the given region by using spherical coordinates.

Explanation

Given:

The region B lies within the sphere x2+y2+z2=4 above the xy-plane and below the cone z=x2+y2 .

Formula used:

The volume of the given region B is, BdV .

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 0 to 2, θ varies from 0 to 2π and to find the limits of ϕ , solve the given equation as below.

z=x2+y2ρcosϕ=ρ2sin2ϕcos2θ+ρ2sin2ϕsin2θρcosϕ=ρsinϕcos2θ+sin2θρcosϕ=ρsinϕ

So, ϕ varies from π3 to π2 . Use the formula mentioned above to find the volume of the given region. Then, by equation (1), the volume of the given region is,

BdV=02ππ3π202dzdydx=02ππ3π204cosϕ(ρ2sinϕ)dρdϕdθ

Integrate with respect to ρ and apply the limit

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

In Exercises 1-6, simplify the expression. 3+(1)2

Calculus: An Applied Approach (MindTap Course List)

Solve the equations in Exercises 126. 10x(x2+1)4(x3+1)510x2(x2+1)5(x3+1)4=0

Finite Mathematics and Applied Calculus (MindTap Course List)

43. Solve for This gives a formula for solving two equations in two variables for.

Mathematical Applications for the Management, Life, and Social Sciences

Using tan2 x = sec2 x − 1, ∫ tan3 x dx =

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

The angle θ at the right has cosine:

Study Guide for Stewart's Multivariable Calculus, 8th

In Exercises 15-22, use the laws of logarithms to solve the equation. logx(116)=2

Finite Mathematics for the Managerial, Life, and Social Sciences 