Midpoint Riemann Sums — Manim Script .... تَقْرِيبُ المَسَاحَةِ بِمُسْتَطِيلَاتِ رِيمَان مُحَاكَاةٌ بَصَرِيَّةٌ لِحِسَابِ التَّكَامُلِ بِطَرِيقَةِ رِيمَان – رَسْمٌ مُتَحَرِّكٌ

This animation shows how numerical integration works by using the midpoint Riemann sum to approximate the area under a curve. The scene presents two synchronized graphs: the top one uses dynamic rectangles that update in real time as the step size changes, while the bottom one displays the exact filled area for comparison. A live numerical value appears and updates continuously as the rectangles shrink, helping viewers understand how the approximation gets closer to the true area. The code is clean, fully commented, and ideal for teachers, students, and content creators who want to demonstrate numerical integration visually and clearly.