The art of accumulation — find areas, volumes, and totals by summing infinitely many infinitesimal pieces.
Antiderivatives
An antiderivative (or indefinite integral) of f(x) is any function F(x) whose derivative is f(x):
∫ f(x) dx = F(x) + C (where F'(x) = f(x))
The "+C" is crucial — there are infinitely many antiderivatives, differing by a constant. Some key antiderivatives:
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
More antiderivatives can be found on the formula sheet.
The Definite Integral
The definite integral computes the signed area between f(x) and the x-axis from a to b:
∫ₐᵇ f(x) dx = lim (n→∞) Σ f(xᵢ)Δx
This is a limit of Riemann sums — rectangles approximating the area. The connection to probability is direct: the probability of a continuous random variable falling in [a,b] is exactly ∫ₐᵇ f(x) dx where f is the probability density function.
The Fundamental Theorem of Calculus
The FTC links differentiation and integration — two seemingly opposite operations are inverses:
Part 1: d/dx [∫ₐˣ f(t) dt] = f(x)
Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x)
The normal distribution, exponential distribution, and every other continuous distribution is defined through integrals.
Integration reaches into every field: physics (work, energy, fluid pressure), engineering (signal processing via Fourier transforms), economics (consumer/producer surplus), and linear algebra (inner products on function spaces). The concept of area under a curve is one of the most widely applied ideas in all of science.