Determine values of A B C and D so that integral30 fxdx Af0

Determine values of A, B, C, and D so that integral^3_0 f(x)dx ~ Af(0) Bf(1) + Cf(2) + Df(3) is exact for all polynomials of degree 3 or less.

Solution

As 1,x, x² ,x³ form a basis for the space of polynomials , we can solve for A,B,C and D by evaluating the integrals fot these four functions.

We obtain A +B+C+D=3

                     B +2C+3D =9/2

                      B +4C + 9D =9

and                B +8C + 27D =81/4

to obtain

                                  A=3/8

                                 B =9/8

                                 C =9/8

and                            D=3/8