The Gaussian Integral – Quick Intuitive Proof

Overview

A fast vertical‑format animation that walks through the classic proof that $\displaystyle I=\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt{\pi}$ by squaring the integral, converting to polar coordinates, and evaluating. The key takeaway is that the Gaussian integral equals $\sqrt{\pi}$.

---

Phases

#	Phase Name	Duration	Description
1	Title & Subtitle	~3 s	Fade‑in the title “The Gaussian Integral” and subtitle “An Intuitive Proof” at the top of the frame.
2	Statement of Integral	~2 s	Write the one‑dimensional integral $I=\int_{-\infty}^{\infty}e^{-x^2}\,dx$ below the subtitle.
3	The Trick: Square It!	~1.5 s	Appear a short cue “The Trick: Square it!” then transform the single integral into $I^2 = \bigl(\int e^{-x^2}dx\bigr)\bigl(\int e^{-y^2}dy\bigr)$.
4	Merge to 2‑D Integral	~2 s	Transform to the double integral $I^2 = \int_{-\infty}^{\infty}\!\int_{-\infty}^{\infty} e^{-(x^2+y^2)}\,dx\,dy$.
5	Switch to Polar Coordinates	~3 s	Fade in text “Switch to Polar Coordinates” with the substitution $x^2+y^2\to r^2$ and $dx\,dy\to r\,dr\,d\theta$.
6	Polar Integral Form	~2 s	Transform to $I^2 = \int_{0}^{2\pi}\!\int_{0}^{\infty} e^{-r^2} r\,dr\,d\theta$.
7	Separate the Integrals	~1.5 s	Split into $\int_{0}^{2\pi}d\theta$ and $\int_{0}^{\infty} e^{-r^2} r\,dr$.
8	Evaluate Radial Part	~1.5 s	Show the antiderivative $-\tfrac12 e^{-r^2}$ and evaluate from 0 to $\infty$.
9	Assemble Result $I^2=\pi$	~1.5 s	Simplify to $I^2 = 2\pi \cdot \tfrac12 = \pi$.
10	Final Reveal	~4 s	Transform to the final equation $I = \sqrt{\pi}$ and draw a highlighted box around it.

---

Layout

┌─────────────────────────────────────────────┐
│                                             │
│                 MAIN (visual)               │
│                                             │
├─────────────────────────────────────────────┤
│   Caption / step label (small, optional)    │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Title, subtitles, all equations, transformation arrows, and the final boxed result.	Takes up the majority of the vertical frame; centered horizontally.
Caption	Brief step label (e.g., “The Trick”, “Polar Coordinates”).	Small text placed just below the main equation during each step; fades out when the step ends.

---

Notes

• Aspect ratio: 9:16 vertical video (TikTok/Shorts) as requested.
• Color palette: Accent = TEAL, Highlight = YELLOW, neutral gray for auxiliary text.
• Font sizes: Title 42 pt, subtitle 24 pt, step cues 28 pt, all MathTex at 34 pt (except final equation at 36 pt).
• Transitions: Primarily FadeIn, TransformMatchingShapes, and FadeOut to keep the pacing brisk.
• Assumptions: No persistent footer text is needed beyond the step cue; the caption area is used only for temporary step labels. All timing estimates are rounded to the nearest half‑second to keep the total length under 30 seconds.
• Single Scene: The entire animation fits within one Scene class.
• No extra text: All information is conveyed visually; no explanatory narration text is included beyond the on‑screen cues.