   Chapter 15.6, Problem 25E

Chapter
Section
Textbook Problem

Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub-boxes of equal size.25. ∭ B cos   ( x y z )   d V , where B   =   { ( x ,   y ,   z ) |   0   ≤   x   ≤ 1 ,   0 ≤   y   ≤   1 ,   0 ≤   z   ≤   1 }

To determine

To estimate: The value of the integral by using the Midpoint Rule for triple integral and also divide the region B into 8 sub-boxes of equal size.

Explanation

Given:

The integral is Bcos(xyz)dV .

The region is B={(x,y,z)|0x1,0y1,0z1} .

Rule used: Midpoint Rule for triple integral

The triple integral of f over the box is, if the limit exists then, Bf(x,y,z)dV=liml,m,ni=1lj=1mk=1nf(xijk,yijk,zijk)ΔV where (xijk,yijk,zijk) is the sample point and ΔV is volume of each sub-box.

Calculation:

To find the volume of each sub-box , each interval of x,y,z must be divided by 2. Therefore,

ΔV=121212=18

Now find the midpoint of each sub-box is

Here divide the region B into 8 sub-boxes with equal sides

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