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Riemann Sums → Definite Integral

受眾: University類別: Math

簡介

Visualizes how Riemann sums approximate the definite integral by increasing the number of rectangles under f(x) = x² from 0 to 3. As n grows from 4 to 32, the approximation converges to the exact area of 9, illustrating the limit definition of the integral.

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Riemann Sums → Definite Integral

Description

Visualizes how Riemann sums approximate the definite integral by increasing the number of rectangles under f(x) = x² from 0 to 3. As n grows from 4 to 32, the approximation converges to the exact area of 9, illustrating the limit definition of the integral.

Phases

# Phase Name Duration Description
1 Setup 5s Title, axes, plot f(x) = x² in blue, label the region [0,3]
2 n = 4 rectangles 7s Draw 4 right-Riemann rectangles, show sum formula, display approximate value
3 n = 8 rectangles 7s Transform to 8 rectangles, update sum value, closer to 9
4 n = 16 rectangles 7s Transform to 16 rectangles, value approaches 9 more
5 n = 32 rectangles 7s Transform to 32 rectangles, nearly fills the area
6 Integral Limit 8s Show limit formula, display ∫₀³ x² dx = 9, shade the exact area
7 Summary 5s Show antiderivative computation [x³/3]₀³ = 9

Layout

+--------------------------------------------------+
|  Title: "Riemann Sums → Definite Integral"       |
+--------------------------------------------------+
|                                                  |
|   Axes with f(x)=x²          | Formula panel    |
|   (left 60%)                 | (right 40%)      |
|                              |                  |
|   Riemann rectangles         | Σ f(xᵢ)Δx        |
|   (gold/orange fill)         |                  |
|   updating as n grows        | ≈ 9.XXX          |
|                              |                  |
|   n label: "n = 4" etc.      | → ∫₀³ x²dx = 9  |
|                              |                  |
+--------------------------------------------------+

Area Descriptions

  • Left 60%: Axes with curve and animated Riemann rectangles
  • Right 40%: Sum formula, running approximation value, final integral result
  • Bottom: n counter and convergence description

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Right Riemann sum: xᵢ = a + i*Δx, rectangle height = f(xᵢ)
  • Exact integral value: ∫₀³ x² dx = [x³/3]₀³ = 27/3 = 9
  • Rectangle fill color: YELLOW with opacity 0.5, border ORANGE
  • Use VGroup of rectangles that transforms between n values