Sphere Path and Limits: Parametric Curve with Definite Integral

This example demonstrates a parametric path moving on a sphere, with dynamic sliders revealing how the path length or related integral approaches a limit as parameters change. Core mathematical concepts include spherical parametric equations, space curves, definite integrals, and the concept of limits, along with subtraction used in interval measurement. Construction highlights: the Angle command creates parameter α, CurveCartesian defines the Cartesian parametric curve, and objects b and a control scale and range, enabling interactive exploration. Suitable for high school to undergraduate courses in calculus and solid geometry, this visualization helps students intuitively grasp projections of parametric curves on spheres, integral approximation, and the geometric meaning of limits.