This is an example of an important integral in the study of normally distributed data. Moreover, it is an integral that cannot be evaluated by the Fundamental Theorem of Calculus using our usual store of "known" functions.

1. Study the summation expression in your worksheet, and explain why it is a Riemann sum approximation to the definite integral.

2. Now evaluate and record the approximating sums for n = 10, 20, 100, 200, and 1000 terms. On the basis of these calculations, give your estimate of the value of the integral. Be careful to include only as many digits in your approximation as you believe to be accurate.

3. Next we look at the Trapezoidal Rule approximation. Explain why the given commands calculate this approximation, and record the approximations for n = 10, 20, and 40 terms.

4. Now the Midpoint Rule approximation -- explain why the given commands calculate this approximation, and record the approximations for n = 10, 20, and 40 terms. Give your best estimate of the integral on the basis of your Trapezoidal and Midpoint approximations.

5. The Simpson's Rule approximation is the weighted sum of Trapezoidal and Midpoint Rule approximations. Identify the weights, and calculate this approximation for n = 20.

6. Finally, compare these results with your helper application's built-in numerical approximation for definite integrals.