Geometrical
Interpretation :
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
‘negative’ areas. ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image002.gif)
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
Principle.
Algorithm to express the infinite series as
definite integral:
Step(i) Express the given series in the
form ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image005.gif)
Step(ii) Then the limit is its sum when
, i.e. ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image007.gif)
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
(a) ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image012.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image013.gif)
(b)![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image014.gif)
where
(as r = 1) and
(as r = pn)
Example.1
Evaluate
S =
.
Solution
Let Sn =
=
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image020.gif)
Hence, S =
Sn = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image021.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image020.gif)
=
=
=
=
.
Example.2
Evaluate
.
Solution
Let Sn =![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image027.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image029.gif)
&⇒ S =
=
= ln6 – ln2 = ln3.
Example.3
Evaluate ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image032.gif)
Solution
Let p = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image033.gif)
Analyzing the
expression with the view of increasing integral value we get the expression in terms
of r as
=
=
,
Put 3
+ 4 = t, \ ![](http://www.quizsolver.com/radix/dth/notif/Definite%20integral%201_files/image037.gif)
Hence p =
.