Numerical integration methods

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, …., x_n = x₀ + nh = b be y₀, y₁, y₂, …., y_n respectively, then Integral a to b f(x) dx = integral x₀ to x₀ + nh = y dx = h[1/2 (y₀ + y_n) + (y₁ + y₂ + ….. + y_n)].

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, …., y_n at n + 1 equidistant points x₀ = a, x₁ = x₀ + h, x₂ = x₀ + 2h, …., x_n = x₀ + nh = b respectively. Then Integral a to b f(x) dx = integral x₀ to x₀ + nh = y dx = h[1/2 (y₀ + y_n) + (y₁ + y₂ + ….. + y_n)] = h/3[(y₀ + y_n) + 4(y₁ + y₃ + ….. + y_n-1) + 2(y₂ + y₄ + ….. + y_n-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.

Its geometric significance is that we replace the graph of the given function by n/2 arcs of second degree polynomials, or parabolas with vertical axes. It is to note here that the interval [a, b] is divided into an even number of subinterval of equal width.