Two-Point Gauss-Legendre Quadrature

In order to illustrate the approach, we first develop the integration formula for the two-point problem. In Newton-Cotes method, the location of the base points is determined, and integration is done based on the values of the function at these base points. This is shown in Fig. 4.2, for the trapezoidal rule that approximates the integral by taking the area under the straight line connecting the function values at the ends of the integration interval. 

consider the case that the restriction of fixed points is withdrawn, and we are able to estimate the integral from the area under a straight line that joins any two points on the curve. By choosing these points in proper positions, a straight line that balances the positive and negative errors can be drawn, as illustrated in Fig. 4.5. Asa result, we obtain an improved estimate of the integral. 

In order to derive the two-point Gauss quadrature, the function y=f(x) is replaced by a linear polynomial and a remainder  

Without loss of generality, the interval [a, b] is changed to [-1,1]. The general transformation equation for converting between x in interval [a, b] and z in interval [c, d] is the following  

For converting to the interval [-1, IJ, this equation becomes Using Eq. (4.93), the transformed integral is given by 

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