Changing the order of integration in a triple integral

 Goal: Change the order of integration in a triple integral over a  general bounded region.

Considerations

Changing the order of integration in double integrals facilitates their computations. In triple integrals over a rectangular box, changing the order of integration doesn't simplify the calculations. However, changing the order of integration in a triple integral over a general bounded region facilitates the computation.

Example

Solution

Let's start by defining the region E and the form in which the triple integral should be expressed once the order of integration is changed:

Let's draw the projections on the three coordinates plane:

Let's redefine E for the change of the order of integration:

The integral becomes:

Let's calculate the given integral by substituting f(x,y,z) by xyz.

= 1/168

Let's calculate the second integral with the order of integration changed.

The answers for both integrals are identical.