Motion of a Point on Continuous Semicircles and Riemann Sums

This applet demonstrates the trajectory patterns of a moving point on continuous semicircles, integrating the concepts of Riemann sums to visually illustrate the geometric meaning of definite integrals. Through dynamic GeoGebra constructions, users can observe the point's position and locus generation as parameters change, gaining a deeper understanding of integral approximation. The applet allows learners to manipulate key parameters such as the number of semicircles, their radii, and the sampling points, watching in real time how upper and lower Riemann sums converge toward the exact area under the curve. By toggling between upper and lower sum approximations, students can visually compare how the choice of sample points affects the estimate, directly connecting this behavior to the formal definition of the definite integral. The construction progressively refines the partition as the user adjusts the step size, showing how increasing the number of subintervals reduces the discrepancy between the upper and lower sums. This hands-on exploration reinforces the fundamental calculus idea that integration is the limit of finite Riemann sums. The applet also highlights the geometric relationship between the arcs of the semicircles and the rectangular approximations used in Riemann sums, making abstract theoretical concepts tangible. Educators can use this tool in classroom settings to guide discussions on limits, convergence, and the conceptual bridge between discrete summation and continuous integration. It serves as an effective visual aid for both introductory calculus courses and advanced review sessions, helping students build intuition before encountering rigorous epsilon-delta definitions.