??? Where did you get "-0.57735" and "0.57735"? I don't see those numbers anywhere in the given data! And f(-0.57735) and f(0.57735) are not known.
The "Two Point Gauss Quadrature" is a very simple numerical integration method when you are given a number of \(\displaystyle (x_i, f(x_i))\) values. It consists of assuming that the function, f(x), is constant from \(\displaystyle x_i\) to \(\displaystyle x_{i+1}\) so that the "area under the curve" is the area of the rectangle with height \(\displaystyle f(x_i)\) and width \(\displaystyle x_{i+1}- x_i\), so area \(\displaystyle f(x_i)(x_{i+1}- x_i)\).
In this example, you have (x, f(x)) equal to (0, 0), (2, 2.1), (3, 2.9), etc. The first few calculations would be 0(2- 0)+ 2.1(3- 2)+ 2.9(3.8- 3), etc.
