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 A_k we choose a point (x_k, y_k, z_k), 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.

 

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

$$\underset{R}{\int \int \int} f(x,y) dA$$

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 displayed on Fig. 8, then:

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

 

  

Figure 8: The union of two regions.

 

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

$$\underset{R}{\int \int \int } dV$$

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.