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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 $\Delta V = \Delta x \Delta y \Delta z$.

  
Figure 7: Partitioning a region in space.
\begin{figure}
\mbox{\epsfig{file=RectangularGrid3d.eps,height=6cm} }
\end{figure}

Define $S_n= \underset{k}{\sum} f(x_k,y_k, z_k) \Delta V_k$. 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

\begin{displaymath}\underset{R}{\int \int \int} f(x,y) dA
\end{displaymath}

Proposition 7.10         
1.
$\underset{R}{\int \int \int} [ f(x,y,z) + g(x,y,z) ] dV =
\underset{R}{\int \int \int} f(x,y,z) dV + \underset{R}{\int \int \int} g(x,y,z) dV$.
2.
$\underset{R}{\int \int \int}\alpha f(x,y,z) dV = \alpha \underset{R}{\int \int \int} f(x,y,z) dV $.
3.
If $\forall (x,y,z) \in R, \; f(x,y,z) \geq 0$, then $\underset{R}{\int \int \int} f(x,y) dV \geq 0$.
4.
If $\forall (x,y,z) \in R, \; f(x,y,z) \leq g(x,y,z)$, then $\underset{R}{\int \int \int} f(x,y,z) dV \leq \underset{R}{\int \int \int} g(x,y,z) dV$.

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:

\begin{displaymath}\underset{R}{\int \int \int } f(x,y,z) dV = \underset{R}{\int...
...nt } f(x,y,z) dV
+ \underset{R}{\int \int \int } f(x,y,z) dV
\end{displaymath}


  
Figure 8: The union of two regions.
\begin{figure}
\mbox{\epsfig{file=2regions.eps,height=6cm} }
\end{figure}

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

\begin{displaymath}\underset{R}{\int \int \int } dV
\end{displaymath}

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