∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 2 Theory

Improper Integrals & Numerical Approximations

Infinite horizons, vertical asymptotes, and Simpson's Rule parabolic fitting

Extend the upper integration limit b toward infinity to discover when tails taper fast enough for total area to stay strictly finite.

1. The p-Integral Convergence Benchmark

The tail integral ∫_1^∞ (1/x^p) dx converges to 1/(p - 1) if p > 1, and diverges to infinity if p ≤ 1. When p = 1 (1/x), the harmonic tail decays too slowly, resulting in infinite area.

∫1∞1xp dx={1p−1p>1∞p≤1\int_1^\infty \frac{1}{x^p}\,dx = \begin{cases} \frac{1}{p-1} & p > 1 \\ \infty & p \le 1 \end{cases}

Quick Reference & Cheat Sheet

Simpson's Parabolic Rule
∫abf(x) dx≈Δx3[f0+4f1+2f2+4f3+⋯+fn]\int_a^b f(x)\,dx \approx \frac{\Delta x}{3}[f_0 + 4f_1 + 2f_2 + 4f_3 + \dots + f_n]
Approximates curves with quadratic polynomials, exact for cubics.