   Chapter 15.6, Problem 53E

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.53. Find the average value of the function f(x, y, z) = xyz over the cube with side length L that lies in the first octant with one vertex at the origin and edges parallel to the coordinate axes.

To determine

To find: The average value of the function f(x,y,z)=xyz over the cube with side L

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 and the function f(x,y,z)=xyz over the cube with length L lies in the first octant and vertex at the origin and the edges are parallel to the coordinate axes.

Calculation:

Formula used:

If g(x) is the function of x , h(y) is the function of y and k(z) is the function of z the,

abababg(x)h(y)k(z)dzdydx=abg(x)dxabh(y)dyabk(z)dz (1)

The region E={(x,y,z)|0xL,0yL,0zL}

V(E) is the volume of E, Therefore

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