   Chapter 15, Problem 34RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.34. ∭ H z 3   x 2   + y 2   +   z 2   d V , where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1

To determine

To calculate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=z3x2+y2+z2 .

The region H is the solid hemisphere centered at origin with radius 1 and lies above the xy-plane.

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:

Since the equation of the sphere is given, solving with spherical coordinates is a wise choice. From the given conditions, ρ varies from 0 to 1, θ varies from 0 to 2π and ϕ varies from 0 to π2

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 112 for x (mentally, if possible). x1=cx+d(c1)

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 107-120, factor each expression completely. 113. 3x2 6x 24

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 1-8, find the derivative of each function. 6.

Mathematical Applications for the Management, Life, and Social Sciences

The curvature of at t = 0 is:

Study Guide for Stewart's Multivariable Calculus, 8th

If f(x) = 2e3x, f(x) = a) 2e3x b) 2xe3x c) 6e3x d) 6xe3x

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