Find the average value of F(x, y, z) over the given region.
-F(x, y, z) = 10x over the cube in the first octant bounded by the coordinate planes and the planes x = 5, y = 5, z = 5
A) 250
B) 1250
C) 25
D) 125

Question

Quizplus · Accepted Answer

The region of integration is a cube with sides of length 5 in the first octant. Therefore, the volume of the cube is V = 5^3 = 125. To find the average value of F over this region, we need to compute the triple integral of F over the region and divide by the volume of the region.

We have

∫∫∫ F(x, y, z) dV = ∫∫∫ 10x dV

The region of integration is

0 ≤ x ≤ 5, 0 ≤ y ≤ 5, 0 ≤ z ≤ 5

So we have

∫∫∫ 10x dV = ∫0^5 ∫0^5 ∫0^5 10x dxdydz 
= ∫0^5 ∫0^5 125 dydz = ∫0^5 625 dz 
= 3125

Therefore, the average value of F over the region is

(1/V)∫∫∫ F(x, y, z) dV = (1/125)(3125) = 25

So the answer is (C).

Find the Average Value of F(x, Y, Z) Over the Given