Calculus×Computing
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Improper integrals

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Integrals over infinite intervals or of unbounded functions, defined as limits: ∫a∞f=lim⁡b→∞∫abf\int_a^\infty f = \lim_{b\to\infty}\int_a^b f. Every probability density over ℝ\R integrates to 1 this way.

Formulas

∫−∞∞e−x2/2 dx=2π\int_{-\infty}^{\infty} e^{-x^2/2}\,\dd x = \sqrt{2\pi}
the Gaussian integral (normalizes the normal distribution)
∫1∞dxxp<∞  ⟺  p>1\int_1^\infty \frac{\dd x}{x^p} < \infty \iff p > 1

Where it shows up in AI

  • Bayesian inference★★★★★fundamentalAI and machine learning

    The evidence p(D)=∫p(D∣θ) p(θ) dθp(D) = \int p(D\mid\theta)\,p(\theta)\,\dd\theta is an integral over the whole parameter space.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

ℒ AI and machine learning

3D Computer graphics

⚙ Robotics and control

∿ Signals, media and vision

⇄ Optimization and systems

⊕ Cryptography and security

ψ Quantum computing and physics

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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