Numerical Integration

Numerical Integration is the process of computing the value of a definite integral, $\int\limits^{b}_{a}f(x)dx,$ when the values of the integrand function, $y=f(x)$ are given at some tabular points. As in the case of Numerical differentiation, here also the integrand is first replaced with an interpolating polynomial, and then the integrating polynomial is integrated to compute the value of the definite integral. This gives us 'quadrature formula' for numerical integration. In the case of equidistant tabular points, either the Newton's formulae or Stirling's formula are used. Otherwise, one uses Lagrange's formula for the interpolating polynomial. We shall consider below the case of equidistant points: $x_0, x_1, \cdots, x_n.$