Areas Under a Normal Curve: Approximating Probabilities with Riemann Sums

This example demonstrates how to approximate the area under a normal (Gaussian) curve using partitioned rectangles or trapezoids, illustrating that the area corresponds to probabilities (e.g., upper/lower tail areas). The key concepts are the normal distribution function, geometric area interpretation of probability, upper and lower sum (bounding approximations), and the Riemann sum idea that approaches the exact value as the partition gets finer—connecting intuition to integration in the limit. Construction typically involves defining the normal curve y=f(x) in GeoGebra, choosing an interval or threshold to mark the desired region, generating subdivision points, drawing the corresponding rectangles/trapezoids for upper/lower sums, and (often) using a slider for the number of subintervals to show convergence of the approximations. Suitable for high school through first-year college courses in probability & statistics and introductory calculus/numerical approximation.