Exercise: Gauss-Legendre on a general interval

Compute $I=\int_1^{1.5} e^{-x^2}\,dx$ with Gauss-Legendre using $n=2$ and $n=3$.

1. Transform [1,1.5] into [-1,1]:

   $$x=\frac{1}{4}y+\frac{5}{4},\qquad dx=\frac{1}{4}dy$$

2. The integral becomes:

   $$I=\frac{1}{4}\int_{-1}^{1}e^{-(y+5)^2/16}\,dy$$

3. With two nodes, ±0.577350 and weights 1:

   $$I_2=0.109400$$

4. With three nodes, 0 and ±0.774597, weights 0.888889 and 0.555556:

   $$I_3=0.109364$$