The integral on the right side of Equation is called an iterated integral.

Usually the brackets are omitted.

means that we first integrate with respect to y from c to d and then with respect to  from a to b. Similarly, the iterated integral

means that we first integrate with respect to  (holding y  fixed) from  to  and then we integrate the resulting function of y with respect to y from  to  .

EXAMPLE 6: Evaluate the iterated integrals. 

Regarding  as a constant, we obtain

Thus the function A in the preceding discussion is given by  in this example. We now integrate this function of x from 0 to 3:

EXAMPLE 7: Evaluate the iterated integrals. 

Here we first integrate with respect to  :