∇
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 5 Theory

Antiderivatives, Riemann Sums & The Fundamental Theorem of Calculus

Accumulating infinitesimal slices into total signed area

Vary partition count N from 3 to 60 to observe rectangle area sums converge toward the exact definite integral ∫_a^b f(x) dx.

1. The Fundamental Theorem of Calculus

FTC connects differentiation and integration as inverse operations: the rate of change of an accumulation function is the integrand itself, and definite integrals equal the net change in any antiderivative.

ddx∫axf(t) dt=f(x),∫abf(x) dx=F(b)−F(a)\frac{d}{dx}\int_a^x f(t)\,dt = f(x), \qquad \int_a^b f(x)\,dx = F(b) - F(a)

Quick Reference & Cheat Sheet

Fundamental Antiderivatives
∫xn dx=xn+1n+1+C  (n≠−1),∫1x dx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \;(n \neq -1), \qquad \int \frac{1}{x}\,dx = \ln|x| + C
Area accumulation under standard power functions.