TABLE OF CONTENTS The definite integral 
is defined as the limits of the sum f( x 1) ? x 1 + f( x 2) ? x 2 + ...... + f( x n) ? x n when n ? ? and each of the lengths ?x 1, ?x 2, ......, ?x n tends to zero. Here ?x 1, ?x 2, ......, ? x n are n sub-intervals into which the range b a has been divided and x 1, x 2, ......, x n are values of x lying respectively in the first, second, ......, nth sub-interval.
A double integral is its counterpart in two dimensions. Let a single-valued and bounded function f( x, y) of two independent variables x, y be defined in a closed region R of the xy-plane. Divide the region R into sub-regions by drawing lines parallel to coordinate axes. Number the rectangles which lie entirely inside the region R, from 1 to n. Let ( x r , y r) be any point inside the rth rectangle whose area is ?A r.
Consider the sum
Let the number of these sub-regions increase indefinitely, such that the largest linear dimension ( i.e., diagonal) of ?A r approaches zero. The limit of the sum...
