Chapter 15.6, Problem 54E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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

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