Chapter 5.5, Problem 31E

Gaussian integration,In an introductory calculuscourse, you may have seen approximation formulas forintegrals of the form ${\displaystyle {\int }_a^{b}f\left(t\right)dt}\approx {\displaystyle \underset{i=1}{\overset{n}{\sum }}{w}_i}f\left(a_i\right)$ ,
where the $a_i$ are equally spaced points on the interval $\left(a,b\right)$ , and the $w_i$ are certain “weights” (Riemann sums, trapezoidal sums, and Simpson’s rule). Gauss has shown that, with the same computational effort, we can get better approximations if we drop the requirement that the $a_i$ be equally spaced. Next, we outline his approach.
Consider the space $P_n$ , with the inner product $\langle f,g\rangle ={\displaystyle {\int }_{-1}^{1}f\left(t\right)g\left(t\right)dt}$ .
Let $f_0,f_1,\mathrm{...},f_n$ , be an orthonormal basis of this space, with degree $\left(f_k\right)=k$ . (To construct such a basis, apply the Gram-Schmidt process to the standard basis $1,t,\mathrm{...},t^n$ .) It can be shown that $f_n$ has n distinct roots $a_1,a_2,\mathrm{...},a_n$ on the interval $\left(-1,1\right)$ . We can find “weights” $w_1,w_2,\mathrm{...},w_n$ such that ${\displaystyle {\int }_{-1}^{1}f\left(t\right)dt}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{w}_{i}f\left(a_i\right)}$ , for all polynomials of degree less than n. See Exercise 4.3.70. In fact, much more is true: This formula holds for all polynomials $f\left(t\right)$ of degree less than 2n.
You are not asked to prove the foregoing assertions for arbitrary n, but work out the case $n=2$ : Find $a_1,a_2$ and $w_i,w_2$ , and show that the formula ${\displaystyle {\int }_{-1}^{1}f\left(t\right)dt}=w_{1}f\left(a_1\right)+w_{2}f\left(a_2\right)$ holds for all cubic polynomials.