Example 3.  Use the Gauss-Legendre quadrature rules for n = 2, 3, 4 and 5 points to compute numerical approximations for   .  

Solution 3.

Find the error for the Gauss-Legendre rules.

It is tedious, but possible to verify the following inequalities for the higher derivatives over the interval  [-1,1].  

We can now establish the  following list of "theoretical" bounds for the errors.

Compare the actual "error" and the "theoretical" bound for the error.

(c) John H. Mathews 2004