There is a wealth of excitement surrounding the recently launched Gemini 3, so I decided to tackle a famous Gaussian integral problem and explore its image creation capabilities.

At first, I just used their default prompt and it’s mind-blowing:

“Create a comic with the title: Getting started with Nano Banana Pro 🍌. Panel 1: A mobile interface on Gemini, rendered in a poetic watercolor-ink style with fine ink outlines and soft, bleeding washes. The interface is friendly and a hand painted with expressive brushwork taps a prominent button labeled “🍌 Create image”. Above the button it should say “Choose your Model” then below there should be a checkbox that says “Thinking with 3 Pro” Muted greys and blues dominate the background. The button has a vivid yellow accent. “ Select the Thinking with 3 Pro model” and tap “Create image” to begin. Panel 2: A cheerful person is depicted adding their selfie from the phone’s camera. The user’s face is drawn with soft outlines and warm pastel colors, while the phone and UI maintain the delicate water-ink aesthetic. Visible paper grain adds to the texture. Panel 3: The person thinks about what to create. In the background, different options are visualized to show what they’re thinking, including — them as a plushie, them with a mohawk hairstyle, and a figurine. These options are clearly rendered behind the person in the same style as the rest of the comic. Panel 4: The person is shown adding a style prompt, with a speech bubble saying “Transform me into a watercolor painting”. The text is integrated into the panel’s watercolor-ink look, and the interaction feels natural and intuitive. Panel 5: The person is seen editing the image by simply typing into Gemini. The scene captures the ease of this interaction, with the final edited image, now in a watercolor style, appearing on the screen. The overall tone is friendly, instructional, and inspiring. It feels like a mini tutorial comic, all conveyed through the specified delicate water-ink illustration style. Make the aspect ratio 16:9.”

Then simple prompt “please do a Gaussian integral and outline steps, then create demo in comic form”, “i meant to ask you to make comic images to demo steps”.

The Gaussian Integral: A Comic Explanation!

Title: “The $\sqrt{\pi}$ Mystery: How to Integrate $e^{-x^2}$”

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Panel 1: The Frustration

• Caption: STEP 1: STARING INTO THE ABYSS.
• Visual: A frustrated mathematician (let’s call him “Dr. Gauss”) stands before a large, insurmountable bell curve (the $e^{-x^2}$ function) on a desolate landscape. He holds a scroll and a quill, looking perplexed.
• Speech Bubble (Dr. Gauss): How do I integrate this?! There’s no antiderivative! I’m trapped in 1D!
• Floating Math (above his head, tangled): $\int x^n, \int u-sub, \int \frac{1}{x}, \int e^x$ (all crossed out with question marks).
• Ground/Function Text: $f(x) = e^{-x^2}$. $I = \int_{-\infty}^{\infty} e^{-x^2} dx = ?$

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Panel 2: The Sneaky Trick – Squaring It

• Caption: STEP 2: A CLEVER DUPLICATION!
• Visual: Dr. Gauss has a sudden “AHA!” moment. He points excitedly. Behind the original bell curve, a ghostly, identical second bell curve appears, rotated 90 degrees along the (imaginary) y-axis.
• Thought Bubble (Dr. Gauss): What if I multiply the integral by ITSELF?! $I^2 = \left(\int e^{-x^2} dx\right) \left(\int e^{-y^2} dy\right)$
• Ground/Function Text: $I^2 = \iint e^{-(x^2+y^2)} dx dy$. (The integral sign is now a double integral).

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Panel 3: Into 2D – The Mountain

• Caption: STEP 3: A NEW DIMENSION, A NEW PROBLEM.
• Visual: The two bell curves have merged to form a beautiful, symmetrical 3D Gaussian “mountain” (like a gentle hill) on a flat, rectangular grid plane. Dr. Gauss scratches his head, holding a clunky, rectangular integration grid in his hand.
• Speech Bubble (Dr. Gauss): A volume problem! $I^2 = \iint e^{-(x^2+y^2)} dx dy$. But this rectangular grid is IMPOSSIBLE to measure with!

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Panel 4: The Polar Revelation

• Caption: STEP 4: A CIRCULAR SOLUTION! POLAR COORDINATES!
• Visual: Dr. Gauss snaps his fingers! The rectangular grid on the plane is replaced by glowing, concentric circles and radial lines (polar coordinates). A small lightbulb appears above his head. He’s smiling.
• Speech Bubble (Dr. Gauss): Change to $r$ and $\theta$! ($x^2+y^2=r^2$) The mountain is ROUND!
• Floating Math/Visual Cues: $x=r\cos\theta$, $y=r\sin\theta$.

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Panel 5: The “r” Factor (Jacobian)

• Caption: STEP 5: THE CRUCIAL ‘$r$’ FACTOR (THE JACOBIAN)!
• Visual: A tiny “mini-Gauss” (the “area element”) is shown on the grid, magnifying a small wedge. Main Dr. Gauss points firmly at a curly, glowing ‘$r$’ symbol that floats.
• Speech Bubble (Main Dr. Gauss): DON’T FORGET THE EXTRA ‘r’! $dx dy \rightarrow r \, dr \, d\theta$.
• Mini-Gauss’s thought: Area Element: $r \, d\theta \, dr$.
• Small floating math: $r$ with musical notes (importance).

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Panel 6: The Solved Integrals!

This is amazing!