Numerical Approximations of Area Under a Curve

This interactive GeoGebra example dynamically demonstrates how to approximate the area under a curve using left-endpoint, right-endpoint, midpoint rectangles, and trapezoids, showing how the approximations converge to the exact integral value as the number of subdivisions increases. The core mathematical concepts include the geometric interpretation of definite integrals, Riemann sums, the trapezoidal rule, and error analysis in numerical integration. Construction highlights: using Textfield and inputBox objects to let users adjust the function, interval endpoints a and b, and the number of subdivisions n; using Sequence and Element to generate partition points and corresponding function values; using the Integral command to compute the exact area; constructing lists for left, right, midpoint, and trapezoidal approximations; and displaying the approximations, exact value, and errors via text objects. This example is suitable for high school seniors or introductory university calculus courses, helping students intuitively grasp the definition of the integral and numerical approximation techniques, and explore the convergence rates of different methods.