my lecturer told me this... when i asked for help... but have no idea what it means, or what he has done

The trick about finding the weights is to consider the Polynomial of

degree 3 of the form

P(x) = f(xj) P_0(x) + P_1(x)f(xj + h/3) + P_2(x)f(xj + 2h/3) P_3(x)f(xj + h)

dont try ax^3 + bx^2 + cx + d, that derivation is many many pages

long, and we require a trick from polynomial interpolation to avoid

complications. It is a trick we used in one of the derivations of

simpsons rule.

Now, we can transform this integral to one over 0 to h, which just

simplifies the calculations to a page or two less. P_i are all of

degree 3 and

P(0) = f(xj)

P(h/3) = f(xj+h/3)

P(2h/3) = f(xj+2h/3)

P(h) = f(xj + h)

this defines the polynomials completely as

P_0(0) = 1 and P_1(0) = P_2(0) = P_3(0) = 0

P_1(h/3) = 1 and P_0(h/3) = P_2(h/3) = P_3(h/3) = 0

P_2(2h/3) = 1 and P_0(2h/3) = P_1(2h/3) = P_3(2h/3) = 0

P_3(h) = 1 and P_0(h) = P_1(h) = P_3(h) = 0

these define the P_i completely since you know each polynomials roots.

Now w_i is the integral of P_j over 0 to h. This is exactly like what

we did for Simpsons rule.