Chapter 15.6, Problem 54E

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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
*E*) is the volume of *E.* For instance, if *ρ* is a density function, then *ρ*_{ave} is the average density of *E*.

**54**. Find the average height of the points in the solid hemisphere *x*^{2} + *y*^{2} + *z*^{2} ≤ 1, *z* ≥ 0.

To determine

**To find:** The average height of the points over solid hemisphere with

Explanation

**Given:**

The average value of the function is,
*E*.

**Calculation:**

*E*, Therefore

From the given observe that
*z* varies from 0 to
*r* varies from 0 to 1 and

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

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