Symbols
∫Integral sign — sum of infinitely thin strips
F(x)Antiderivative — F'(x) = f(x)
a, bLower and upper limits of integration
nNumber of Riemann rectangles
ΔxWidth of each rectangle = (b−a)/n
Key Theorems
Fundamental Theorem of Calculus
The definite integral equals F(b)−F(a) where F is any antiderivative of f. This connects differentiation and integration.
Riemann Sum Definition
As n→∞ the sum of rectangle areas converges to the exact integral. Left/Right/Midpoint/Trapezoid have different convergence rates.
Trapezoidal Rule
Approximates each strip as a trapezoid. More accurate than left/right Riemann for smooth functions — O(1/n²) error.