Constants for a quadrature formula

[JellyFish]

This is a problem I have been working on:
Find the constants C[sub:3ka9jcb7]0[/sub:3ka9jcb7], x[sub:3ka9jcb7]0[/sub:3ka9jcb7], and x[sub:3ka9jcb7]0[/sub:3ka9jcb7] so that the quadrature formula:
integral from 0 to 1 of f(x) = (1/2)f(x[sub:3ka9jcb7]0[/sub:3ka9jcb7]) + C[sub:3ka9jcb7]1[/sub:3ka9jcb7]f(x[sub:3ka9jcb7]1[/sub:3ka9jcb7])

gives exact results for all polynomials of degree 3 or less.

Am I right to assume that C[sub:3ka9jcb7]0[/sub:3ka9jcb7] = 1/2?

So far I have used a method in the text book in which you do the
intgral from 0 to 1 of a[sub:3ka9jcb7]0[/sub:3ka9jcb7]+a[sub:3ka9jcb7]1[/sub:3ka9jcb7]x +a[sub:3ka9jcb7]2[/sub:3ka9jcb7]x[sup:3ka9jcb7]2[/sup:3ka9jcb7] +a[sub:3ka9jcb7]3[/sub:3ka9jcb7]x[sup:3ka9jcb7]3[/sup:3ka9jcb7]

By evaluating each integral and then setting each part equal to the constants C[sub:3ka9jcb7]0[/sub:3ka9jcb7] and C[sub:3ka9jcb7]1[/sub:3ka9jcb7] I have

1/2 + C[sub:3ka9jcb7]1[/sub:3ka9jcb7] = 1, so then C[sub:3ka9jcb7]1[/sub:3ka9jcb7] = 1/2

(1/2)x[sub:3ka9jcb7]0[/sub:3ka9jcb7] + (1/2)x[sub:3ka9jcb7]1[/sub:3ka9jcb7] = 1/2

(1/2)(x[sub:3ka9jcb7]0[/sub:3ka9jcb7])[sup:3ka9jcb7]2[/sup:3ka9jcb7] + (1/2)(x[sub:3ka9jcb7]1[/sub:3ka9jcb7])[sup:3ka9jcb7]2[/sup:3ka9jcb7] = 1/3

(1/2)(x[sub:3ka9jcb7]0[/sub:3ka9jcb7])[sup:3ka9jcb7]3[/sup:3ka9jcb7] + (1/2)(x[sub:3ka9jcb7]1[/sub:3ka9jcb7])[sup:3ka9jcb7]3[/sup:3ka9jcb7] = 1/4

After this I get stuck with sloving for the two x's. Also does this whole method seem correct?

 

[DrMike]

1/2 + C[sub:1fkizgqp]1[/sub:1fkizgqp] = 1, so then C[sub:1fkizgqp]1[/sub:1fkizgqp] = 1/2

This is fine.

(1/2)x[sub:1fkizgqp]0[/sub:1fkizgqp] + (1/2)x[sub:1fkizgqp]1[/sub:1fkizgqp] = 1/2

So x[sum]0[/sub]=1-x[sub:1fkizgqp]1[/sub:1fkizgqp], which you substitute into

(1/2)(x[sub:1fkizgqp]0[/sub:1fkizgqp])[sup:1fkizgqp]2[/sup:1fkizgqp] + (1/2)(x[sub:1fkizgqp]1[/sub:1fkizgqp])[sup:1fkizgqp]2[/sup:1fkizgqp] = 1/3

to get a quadratic equation for x[sub:1fkizgqp]1[/sub:1fkizgqp], and solve to find x[sub:1fkizgqp]1[/sub:1fkizgqp] and x[sub:1fkizgqp]0[/sub:1fkizgqp]. Substitute these into

(1/2)(x[sub:1fkizgqp]0[/sub:1fkizgqp])[sup:1fkizgqp]3[/sup:1fkizgqp] + (1/2)(x[sub:1fkizgqp]1[/sub:1fkizgqp])[sup:1fkizgqp]3[/sup:1fkizgqp] = 1/4

and if the right hand side equals the left hend side, you are done. Otherwise, there is no solution.