Fubini’s Theorem for Triple Integrals 

Fubini’s Theorem for Triple Integrals If  is continuous on the rectangular , then

EXAMPLE 1: Evaluate the triple integral , where is the rectangular box given by 

We could use any of the six possible orders of integration. If we choose to integrate with respect to , then , and then , we obtain 

Now we define the triple integral over a general bounded region E in three-dimensional space  (a solid) by much the same procedure that we used for double integrals. We enclose E in a box B of the type given by . Then we define F so that it agrees with f on E but is 0 for points in B that are outside E. By definition,

We restrict our attention to continuous functions   and to certain simple types of regions. A solid region E is said to be of type 1 if it lies between the graphs of two continuous functions of  and , that is,  By the same sort of argument that led to, it can be shown that if  is a type 1 region given by Equation , then 

In particular, if the projection D of E onto the xy-plane is a type I plane region, then 

and Equation becomes

If, on the other hand, D is a type II plane region, then

and Equation becomes

EXAMPLE 2: Evaluate , where 

EXAMPLE 3: Evaluate , where

Although this integral could be written as

it’s easier to convert to polar coordinates in the xy-plane:  . This gives