Question 8.7: To test our claim about the accuracy of these methods, use P...
To test our claim about the accuracy of these methods, use Program 8.4 to calculate the integral
I=(m+1)∫^{1}_{0} x^m dx (8.3.22)
which we know has the value I = 1 for any positive integer m.
Table 8.1 shows the results up to m = 6. There are two important things to note here. First, as we move down a given column in the table, once we exceed the order of the polynomial used for the interpolation in a particular method, then the error increases monotonically with m. This occurs because the higher-order polynomials are increasingly curved, so the values of their derivatives, which appear in the equation for Rn in Eq. (8.2.69), are increasing. Second, note that the error for n = 2 and n = 3 begins at m = 4, and the error for n = 4 and n = 5 begins at m = 6. This is the manifestation of the zero value of the odd integral in Eq. (8.3.15) for n = 2 and a similar integral at the fifth power for n = 4
R_n=\frac{f^{n+1}(ξ ))}{(n+1)!}h^{n+1} α(α − 1)(α − 2)··· (α − n) (8.2.69)
\int_{0}^{2}{\alpha (\alpha -1)(\alpha -2)d\alpha }=\left[\frac{\alpha^4 }{4}-\alpha ^3+\alpha ^2 \right]_0^2 (8.3.15)