Please answer! Gaussian Quadrature?

Question

I have a question already up...I didn't mean for people to think they should do it for me, I'm very confused about how to use the Gaussian method on this one. I would just like someone to help me point out how to begin doing this problem. The problem is labeled something like Teacher defines polynomials by T0, T1, T2...

Answer 1

n numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1, by a suitable choice of the n points xi and n weights wi. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as

\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)

It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.

Another stable quadrature rule, of accuracy comparable to Gaussian quadrature but with some computational advantages, is Clenshaw-Curtis quadrature.

Answer 2

The fact that people can actually learn this level of math is amazing and impressive to me. Wish you luck in finding your answer.