Example 1.  Use the Gauss-Legendre quadrature rules for n = 2, 3, 4 and 5 points to compute numerical approximations for    .  

Solution 1.

Find the error for the Gauss-Legendre rules.

Since we are using the example , the higher derivative are all bounded by 1 and we can use the inequalities  .
We have 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