Introduction
to numerical integration: First here we will understand the
concept What is Numerical Integration As we know in integration that if ɸ(x) is
primitive of f(x) defined on [a, b], then integral a to b f(x) dx = ɸ(b) - ɸ (a).
In most of the practical problems we are given a set of numerical values of the
function f(x) corresponding to some values of x and in some cases either the
primitive does not exist or it cannot be easily determined by elementary means;
as a result the computation of the definite integral by the above formula may
not be possible. In such circumstances, the numerical methods for integration
are needed.
Numerical integration is
the process of computing the value of a definite integral when we are given a
set of numerical values of the inte-grand f(x) corresponding to some values of
the independent variable x. And also Methods
of Numerical Integration are The
trapezoidal rule: Let y = f(x) be a function defined on [a, b] which is divided
into n equal su-intervals each of width h so that b – a = nh. Let the values of
f(x) for (n + 1) equidistant arguments x0 = a, x1 = x0
+ h, x2 = x0 + 2h, …., xn = x0 + nh
= b be y0, y1, y2, …., yn
respectively, then Integral a to b f(x) dx = integral x0 to x0
+ nh = y dx = h[1/2 (y0 + yn) + (y1 + y2
+ ….. + yn)].
This rule is known as trapezoidal rule. In the
derivation of this formula it is assumed that the y is a linear function of x
i.e., the equation of the curve is the form y = a + bx and second rule is Simpson’s
one-third rule: Let y = f(x) be a function defined on [a, b] which is divided
into n (an even number) equal parts each of width h so that b – a = nh. Suppose
the function y = f(x) attains values y0, y1, y2,
…., yn at n + 1 equidistant points x0 = a, x1
= x0 + h, x2 = x0 + 2h, …., xn = x0
+ nh = b respectively. Then Integral a to b f(x) dx = integral x0 to
x0 + nh = y dx = h[1/2 (y0 + yn) + (y1
+ y2 + ….. + yn)] = h/3[(y0 + yn) +
4(y1 + y3 + ….. + yn-1) + 2(y2 + y4
+ ….. + yn-2)] = (one –third of the distance between two consecutive
ordinates) × [(sum of the extreme ordinates) + 4(sum of odd ordinates) + 2(sum
of even ordinates)] this formula is known as Simpson’s one-third rule.
