Triple integrals over a bounded region

 We expand the definition of the triple integrals to calculate the triple integral of a function f(x,y,z) over a general bounded region E in R³. We start first by projecting this region on the xy plane.

Let D be the bounded region that is a projection of E. We define E as follows:

The  2 functions are shown in the figure above.

The following theorem defines the triple integral of the function f(x, y, z) over the region E.

Theorem

The triple integral of a continuous function f(x,y,z) over a general three-dimensional region

in R³, where D is the projection of E onto the xy-plane is

The calculation of the triple integral depends on which plane is the region D. is the region. In the theorem above the calculation is based on the region D located in the xy plane.. Now we consider the region D in the xz plane and two functions  y = u₁(x,y) and y = u₂(x.y) such that u₁(x,y) ≤ u₂(x,y) for all (x, z) in D. The region E can be defined as:

The region D in any plane can be of type I or type II as mentioned in double integrals over general

 regions. If D is of type I (figure below). Then the region E is defined as follows :                           :

                    

 

If D in the xy plane is of type II as shown in the figure below then the region E is define as follows:

Example                                                                                                                                                      

Solution

 

The following figures show the solid tetrahedron E and its projection on the xy plane. 

   

                       

 

             

This result is obtained by considering 5x-3y as a constant and then calculating  the integral                .

                 

The above result is obtained by differentiating with respect to y and factoring  .We may not factorize and obtain the same result