Example 4.  Use cubic spline quadrature to compute a numerical approximation to the integral  .  
Use the tolerances .  Compare with the analytic or "true value" of the integral.

Solution 4.

4 (a). Plot the function over the interval  [0, 2].

4 (b). Construct the cubic spline for 11 nodes and use it for quadrature.

4 (c). Construct the cubic spline for 21 nodes and use it for quadrature.

4 (d). Construct the cubic spline for 41 nodes and use it for quadrature.

4 (e). Construct the cubic spline for 41 nodes and use it for quadrature.

4 (f). Compare the results from parts b-d.

m sample points
11
21
41
81

4 (g). Use Mathematica to find the analytic solution to the integral, i.e. the "true value" of the integral.

4 (h). How close did our last numerical approximation using Romberg integration come to the "true value" of the integral.

(c) John H. Mathews 2004