Fundamental Theorem of Calculus – Visual Deduction

Overview

A concise visual proof of the Fundamental Theorem of Calculus, showing how the derivative of the integral function $F(x)=\int_{a}^{x} f(t)\\,dt$ equals the original integrand $f(x)$. The animation highlights the geometric meaning of accumulation and the limit of a Riemann sum, culminating in the equality $F'(x)=f(x)$.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title "Fundamental Theorem of Calculus" fades in; a simple coordinate plane appears.
2	Accumulation Function	~6s	Define $F(x)=\int_{a}^{x} f(t)\\,dt$. Show a curve $y=f(t)$ and a shaded area from $a$ to a moving point $x$. The moving point slides right, dynamically updating the shaded area.
3	Riemann Approximation	~7s	At a specific $x$, replace the smooth area with a thin vertical rectangle of width $\Delta x$ and height $f(x)$. Illustrate the limit process as $\Delta x\ o 0$.
4	Derivative Emerges	~6s	Show the change in $F$ for a small increment $\Delta x$: $\frac{F(x+\Delta x)-F(x)}{\Delta x}$. Animate the quotient simplifying to $f(x)$ as the rectangle height approaches the curve.
5	Formal Statement	~4s	Present the theorem statement $\displaystyle \frac{d}{dx}\int_{a}^{x} f(t)\\,dt = f(x)$ with a brief fade‑in of the formula.
6	Outro	~2s	Fade out leaving only the theorem formula centered.

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Layout

┌─────────────────────────────────────┐
│               TOP AREA               │
├───────────────────────┬─────────────┤
│                       │             │
│        LEFT AREA      │ RIGHT AREA  │
│   (main graph, area  │ (optional   │
│    shading, rectangles)  labels) │
│                       │             │
├───────────────────────┴─────────────┤
│               BOTTOM AREA            │
│   Small equations / captions          │
└─────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Fundamental Theorem of Calculus" (Phase 1)	Fades in, stays for first 3 s, then fades out
Left	Coordinate plane with curve $y=f(t)$, shaded accumulation area, moving point, and rectangle visualizations	Primary visual focus throughout phases 2‑4
Right	Brief textual labels such as "Accumulated area", "Δx rectangle", and "Limit" when they first appear	Appear with a soft fade, disappear after their phase
Bottom	Small equations: definition of $F(x)$, difference quotient, final theorem statement	Appear only in phases 2, 4, and 5 respectively

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Notes

• Keep the total runtime under 30 seconds (≈28 s). All transitions are simple fades or linear motions.
• No spoken narration; visual cues (arrows, highlighting) replace text where possible.
• The function $f(t)$ can be a simple positive curve, e.g., $f(t)=\sin t + 2$, to ensure the area is always visible.
• The moving point and rectangle should have a distinct color (e.g., orange) contrasting with the curve (blue) and shaded area (light blue).
• The limit visualization (Δx → 0) uses a shrinking rectangle width while keeping height fixed, emphasizing the derivative.
• Ensure the final theorem formula stays on screen for at least 2 seconds before fade‑out.