Area Under the Curve – Riemann Sum to Integral

Overview

This animation visually demonstrates how the area under a curve can be approximated by rectangles. Starting with a few thick rectangles, they progressively become thinner until the approximation converges to the exact area under the curve – illustrating the concept of the definite integral.

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Phases

#	Phase Name	Duration	Description
1	Setup	~3s	Display coordinate axes and plot the function curve (e.g., $f(x) = x^2$ from $x=0$ to $1$). A label "Area Under the Curve" fades in at the top.
2	Coarse Approximation	~4s	Show $4$ rectangles under the curve, each with height equal to the function value at the left endpoint. Rectangles are semi‑transparent blue. A caption appears: "4 rectangles – rough estimate".
3	Refining Rectangles	~10s	Repeatedly double the number of rectangles (8, 16, 32, 64…) with smooth morphing/transitions. Each step shows the rectangles getting thinner and filling the area more accurately. The caption updates to show the current number of rectangles.
4	Exact Area (Limit)	~3s	After the last step (e.g., 128 rectangles), fade the rectangles into a solid shaded area under the curve. Display the definite integral notation $\int_0^1 x^2\,dx = \frac{1}{3}$ and the label "Exact Area". Briefly hold.

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Layout

┌─────────────────────────────────────────────┐
│            Top Label                        │
├─────────────────────────────────────────────┤
│                                             │
│       MAIN: Graph + Rectangles              │
│                                             │
├─────────────────────────────────────────────┤
│            Bottom Caption                   │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top Label	Text "Area Under the Curve" (persistent)	Disappears after the final phase; replaced by the integral result.
Main	Cartesian plane with axes, the function curve, and approximating rectangles	Rectangles are filled with a semi‑transparent color (e.g., blue with opacity 0.4). The curve is drawn in a contrasting color (e.g., white or yellow). Axes are standard with tick marks at integer multiples.
Bottom Caption	Dynamic text showing the number of rectangles and/or the Riemann sum value	In Phase 2: "4 rectangles". In Phase 3: updates to "8 rectangles", "16 rectangles", etc. In Phase 4: replace with "Exact Area" and the integral equation.

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Notes

• Use the function $f(x) = x^2$ on the interval $[0,1]$ for simplicity and a clear visual result. The left‑endpoint Riemann sum is used.
• Transitions between rectangle counts should be smooth (e.g., animating the removal of old rectangles and adding new ones with Transform or Create/FadeOut). Prefer Transform to keep the visual flow.
• The total animation length should be around 20 seconds (adjust durations if needed to keep under 30 seconds).
• At the end, the shaded area can be created by filling the region under the curve with a gradient, and the integral notation should appear with a simple fade-in.
• No audio, interactivity, or complex rendering parameters are needed.