Next: Evaluation of a triple Up: Triple integrals. Previous: Triple integrals.

## Triple integrals: definition and first properties.

Let f(x,y,z) be a function defined on a domain R in the three-dimensional space. We divide the domain R with a network of lines parallel to the coordinate axes, and we number the little boxes A1, A2, and so on. On each little box Ak we choose a point (xk, yk, zk), as on Fig  7. The volume of such a little box is denoted .

Define . If this sum has a finite limit when we refine indefinitely the network, then this limit is called the triple integral of f over R and is denoted

Proposition 7.10
1.
.
2.
.
3.
If , then .
4.
If , then .

Proposition 7.11   Suppose that the region R is the union of two regions R1 and R2, the border line being a smooth surface, as dispayed on Fig.  8, then:

If the function f is constant over R and f=1, then the triple integral

computes the volume of R.

In Calculus I, we learnt how to compute the volume of a solid of revolution; here we have a more general tool, which enables us to computes volumes for strange'' regions of the space.

Next: Evaluation of a triple Up: Triple integrals. Previous: Triple integrals.
Noah Dana-Picard
2001-05-30