   Chapter 15.6, Problem 54E

Chapter
Section
Textbook Problem

The average value of a function f (x, y, z) over a solid region E is defined to be f a v e   =   1 V ( E ) ∭ E f ​ ( x , y ,   z )   d V where V(E) is the volume of E. For instance, if ρ is a density function, then ρave is the average density of E.54. Find the average height of the points in the solid hemisphere x2 + y2 + z2 ≤ 1, z ≥ 0.

To determine

To find: The average height of the points over solid hemisphere with x2+y2+z21 and z0

Explanation

Given:

The average value of the function is, fave=1V(E)Ef(x,y,z)dV where V(E) is the volume of E.

Calculation:

V(E) is the volume of E, Therefore

volume of hemisphere=23πr3=23π(1)3[x2+y2+z21]=23π

From the given observe that x2+y2+z21 but it is in polar form. So let x=rcosθ and y=rsinθ it implies z1r2 . Therefore z varies from 0 to 1r2 and  r varies from 0 to 1 and θ varies from 0 to 2π .

Ef(x,y,z)dV=02π0101r2zdzdrdθ

Integrate the above integral with respect to z and apply the limit of it.

Ef(x,y,z)dV=02π01[z22]01r2rdrdθ=1202π01[(1r2)2(0)2

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

Absolute Value Evaluate each expression. 70. (a) |212| (b) |111|

Precalculus: Mathematics for Calculus (Standalone Book)

In Exercises 1124, find the indicated limits, if they exist. 12. limx1 (x2 + 1)

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

Evaluate limx[xx2ln(1+xx)].

Single Variable Calculus: Early Transcendentals

True or False: converges.

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

Given: m1=8(x+2)m3=5x3m5=5(x+1)2 Find: x

Elementary Geometry for College Students 