Let f(x) be a function
defined on a closed interval [a, b]. Then definite integral represents the
algebraic sum of the areas of the region bounded by the curve y = f (x) and the
x-axis between the lines x = a and x = b. All the regions lying above the
x-axis have ‘positive’ areas whereas those lying below the x-axis have
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]
i.e., . The converse is also
true i.e., if we have an infinite series of the above form, it can be expressed
as a definite integral.
The method to evaluate the integral, as
limit of the sum of an infinite series is known as Integration by First
Algorithm to express the infinite series as
Step(i) Express the given series in the
Step(ii) Then the limit is its sum when , i.e.
Step(iii) Replace by x and by dx and by the sign of .
Step(iv) The lower and the upper limits of
integration are the limiting values of for the first and
the last term of r respectively.
Some particular cases of
the above are
where (as r = 1) and
(as r = pn)
S = .
Let Sn = = =
Hence, S = Sn =
= = = = .
Let Sn =
&⇒ S = = = ln6 – ln2 = ln3.
Let p =
expression with the view of increasing integral value we get the expression in terms
of r as
Put 3+ 4 = t, \
Hence p = .