In this topic we discuss a repeated process of integration of a function of two and three variables reffered to as:
The double integral of a function f (x,y) over thw region D in R2 is denoted by,
Let f (x,y) be a continuous function in R2 defined on a closed rectangle;
For any fixed x ∈ [a,b] consider the integral
Similarly we can define another;
The value of integral depends on the x and we get a new function of x. This can be integrated depends on x and we get;
This is called Iterated Integrals.
For continuous function f (x,y) we have;
If f (x,y) is continuous on a bounded region S and S is given by,
are two continuous functions on [a,b] then,
Then, the iterated integral in the R.H.S is also written in the form of
If S can not be written in neither of the above two forms we divide S into finite numbers of subregion such that each of the subregion can be represented in one of the above forms and we get the double integral over S by adding the integrals over these subregion.