Quadrature Using Legendre Polynomials

I am working on studying for my last Qualifying Exam and ran across a problem that has stumped me. I have seen someone else's work for this, but I am pretty it is wrong. Anyway...

Quote:

Let

be a smooth function and consider the integral

and consider

that interpolates

at the roots of the Legendre polynomial of degree n.

Use

to derive a quadrature method for the integral

The solution I have seen is as follows

Quote:

Let

be the

distinct roots of

(the Legendre Polynomial of degree

)

Define

So my issue is that he didn't really use Legendre polynomials. Instead he bipassed the Legendre polynomials in favor of a Lagrange polynomial that interpolates at the Legendre roots. Maybe I am missing something or just confused. Could someone help me either understand the work above or help me with the problem? Thanks in advanced.