Two Thirds: Visualizing Area and Integration

① This example demonstrates the fraction 'two thirds' through a geometric construction involving a rectangle, circles, sectors, and vectors, with interactive bilingual text and buttons. ② Core mathematical concepts include area calculation, definite/indefinite integrals as tools for area, linear programming feasible regions, and the partitioning of areas into rectangles and sectors. ③ Construction details: Uses GeoGebra commands such as Circle, Sequence, and Vector to create circles, point sequences, and vectors, generating multiple objects (a, b, O, c, A, Points, Segments, Sectors, etc.), with text objects (text2, text3, etc.) and buttons (button1, button2) for language switching and dynamic display. ④ Suitable for upper secondary or pre-university students to understand the relationship between fractions and geometric areas, and the role of integration in area computation, also serving as an interactive bilingual teaching resource.