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Explain how to construct a Riemann sum for a function of three variables over a rectangular solid.

Short Answer

Expert verified

The triple integral of fover Ris given by the following Riemann Sum:

role="math" localid="1650345492114" Rfx,y,zdV=liml,m,ni=1lj=1mk=1nfxi,yj,zkV

provided that this limit exists.

Step by step solution

01

Step 1. Given information

We need to explain how to construct a Riemann sum for a function of

three variables over a rectangular solid.

02

Step 2.Explanation for constructing a Riemann sum for a function of three variables over a rectangular solid.

Let a1<a2,b1<b2and c1<c2be real numbers and Rbe the rectangular solid defined by,

R={x,y,z|a1xa2,b1yb2and c1zc2}

Let fx,y,zbe a function defined on R.

The triple integral of fover Ris given by the following Riemann sum:

Rfx,y,zdV=limmaxxi,yj,zk0i=1lj=1mk=1nfx*i,y*j,z*kVijk=liml,m,ni=1lj=1mk=1nfxi,yj,zkV

provided that this limit exists.

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