We know that when solving double integrals, we divide the two dimensional region into very small rectangles. Then we multiply the area of the rectangles with the value of the function at that point. Then we sum up these areas and then apply the following limit: lim (size of rectangle -> 0). Doing all that gives us the double integral of the said function.
Let us now try extending this concept to three dimensions. Just like in double integrals we had some region in a plane (say the xy plane), in triple integral we will consider a solid in space (i.e. xyz space). Just like in double integrals we had split the region into rectangles, in solving triple integrals we break down the solid into numerous rectangular solids (or cuboids). Extending further on same lines, just like how in double integrals we multiplied the value of the function by the area of the respective rectangle, in a triple integral example, we would multiply the value of the function at each of the point by the volume of the rectangular solid at that point. Instead of the limit of size of rectangle tending to zero, in case of triple integrals we have the limit as the volume of rectangle tending to zero.
With that we come to the definition of a triple integral. Which is like this: Triple integral is defined as the limit of the sum of product of volume of rectangular solids with the value of the function.
We call the double integral as an equivalent to double iterated integral. In the same way we can understand the triple integral as a triple iterated integral.
Symbolically the definition of a triple integral can be stated as follows:
Consider a function f(x,y,z). It is of three variables. It is continuous over any solid S. Then the triple integral of this function over the solid S can be symbolically stated as:
where the sum is taken over all the rectangular solids that are contained in the solid S. The limit is for the side length of the rectangles.
The above definition of triple integrals is useful only when we are given a set of data in the form of a table of values of volume and value of function. When a function is defined symbolically, then we use the Fubini’s theorem to solve triple integral examples.
