Definite integral as limit of a sum:

An alternative way of describing  is that the definite integral is a limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b] that is  where h = b-a/n. The converse is true that is, if we have an infinite series of the above form, it is expressed as a definite integral.

Method to express infinite series as definite integral:

(i) Express given series in the form 

(ii) Then limit is its sum when n->∞, i.e. 

(iii) Replace r/n by x and 1/n by dx and by the sign of ∫.

(iv) The lower and upper limit of integration are limiting values of r/n for the 1^st and the last term of r respectively.

Some of the particular cases of the above are given below

Illustration:  Show that

Solution:             (A) Let  I =  

                              (B)            

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