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.

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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

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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

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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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