My first triple improper integral via spherical coordinates

Overview

blackpenredpen evaluates the integral of √(x²+y²+z²) e^{-(x²+y²+z²)} over all of ℝ³ by switching to spherical coordinates. Using the Jacobian ρ² sin φ and full-space bounds, the integral separates into radial, azimuthal, and polar factors whose product is 2π.

Key takeaways

For a radial integrand over all of ℝ³, spherical coordinates replace x²+y²+z² with ρ² and simplify the integral substantially.
The spherical-coordinate Jacobian contributes ρ² sin φ; combined with the original distance factor ρ, it produces the radial term ρ³.
The bounds ρ ∈ [0, ∞), θ ∈ [0, 2π], and φ ∈ [0, π] cover three-dimensional space without double-counting.
The integral separates into independent radial and angular factors because the transformed integrand is ρ³e^{-ρ²} sin φ.
The radial integral ∫₀∞ ρ³e^{-ρ²}dρ equals 1/2, while the two angular integrals equal 2π and 2, yielding 2π overall.

Chapters

0:00 Recognizing a Three-Dimensional Gaussian Integral

The target is the triple improper integral of √(x²+y²+z²) e^{-(x²+y²+z²)} over x, y, and z from −∞ to ∞.
The familiar two-dimensional Gaussian-integral strategy uses polar coordinates; the three-dimensional version requires spherical coordinates.
blackpenredpen introduces spherical variables ρ, θ, and φ as distance from the origin and two successive rotations.

2:20 Converting the Integrand and Full-Space Bounds to Spherical Coordinates

The distance identity x²+y²+z² = ρ² turns the square-root factor into ρ and the exponential into e^{-ρ²}.
The spherical volume element is dV = ρ² sin φ dρ dθ dφ, giving a combined integrand ρ³e^{-ρ²} sin φ.
To cover all of ℝ³ once, use 0 ≤ ρ < ∞, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π.

5:55 Separating the Integrals and Evaluating the Result 2π

The spherical integral factors into ∫₀∞ ρ³e^{-ρ²}dρ, ∫₀²π dθ, and ∫₀π sin φ dφ.
Integration by parts gives the radial factor 1/2; exponential decay makes the terms at infinity vanish.
The angular factors are 2π and 2, so their product with 1/2 gives the final value 2π.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.