Chapter 4, Problem 6PS

(a)

To determine

To calculate: Using two point Gaussian Quadrature approximation ${\displaystyle \underset{-1}{\overset{1}{\int }}f\left(x\right)dx}=f\left(-\frac{1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right)$, approximate the integration and also the error of the approximation of the integral

$I={\displaystyle \underset{-1}{\overset{1}{\int }}\cos \left(x\right)dx}$

(b)

To determine

To calculate: Using two point Gaussian Quadrature approximation ${\displaystyle \underset{-1}{\overset{1}{\int }}f\left(x\right)dx}=f\left(-\frac{1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right)$ approximate the integration and also the error of the approximation of the integral

$I={\displaystyle \underset{-1}{\overset{1}{\int }}\frac{1}{1+x^2}dx}$

(c)

To determine

To prove: That the Two-Point Gaussian Quadrature Approximation ${\displaystyle \underset{-1}{\overset{1}{\int }}f\left(x\right)dx}=f\left(-\frac{1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right)$ is exact for all polynomials of degree 3 or less