Integration Counter-examples: Witch's Hat Functions

This example demonstrates the incorrectness of swapping integral and limit symbols using Witch's hat functions. It involves core math concepts like definite integral, function graph, integral calculus, limits, piecewise functions, fractal geometry, incircle, trapezoid, sphere, and distributions. Constructed with GeoGebra commands such as Polygon and PointIn. Suitable for high school students and educators to understand integral properties and limit applications.