The integral computes the exact total accumulation of a quantity — most famously, the exact area between a curve and the x-axis. Defined as a limit of increasingly fine approximations, the integral is connected to the derivative by the Fundamental Theorem of Calculus, one of the most important results in all of mathematics.
Finding the area of a rectangle is easy: length times width. But what about the area under a curved line — like the area under a hill-shaped graph? You can't just multiply two numbers together, because the height keeps changing.
This idea — the "method of exhaustion" — was pioneered by Archimedes around 250 BCE to find areas and volumes of curved shapes, over 1,900 years before integral calculus was formally developed by Newton and Leibniz.
Divide the interval [a,b] into n subintervals. In each subinterval, pick a sample point and multiply the function's value there by the subinterval's width — this gives a "Riemann sum," approximating the total area with a collection of thin rectangles. As n → ∞ and the subintervals shrink to zero width, if the Riemann sums converge to a single value regardless of how the sample points are chosen, that value is the definite integral:
This theorem connects derivatives and integrals — the two halves of calculus — showing they are inverse operations:
Part 1: If F(x) = ∫ₐˣ f(t) dt, then F′(x) = f(x). (The integral, differentiated, gives back the original function.)
Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f (i.e. F′ = f). This turns the hard problem of computing an integral (as a limit of sums) into the often-easier problem of finding an antiderivative.
The antiderivative (or indefinite integral) ∫f(x)dx = F(x) + C represents the family of all functions whose derivative is f(x). The "+C" (arbitrary constant) reflects that adding any constant to a function doesn't change its derivative.
| Function | Integral |
|---|---|
| xⁿ (n≠−1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| cos(x) | sin(x) + C |
| sin(x) | −cos(x) + C |
Unlike differentiation (which is largely mechanical), integration often requires cleverness: substitution (reversing the chain rule), integration by parts (reversing the product rule: ∫u dv = uv − ∫v du), and partial fractions (breaking rational functions into simpler pieces).
The Riemann integral fails for some important functions — most famously the Dirichlet function (1 on rationals, 0 on irrationals), which is not Riemann integrable because it oscillates too wildly for the sums to converge to a common value. This and other issues motivated a more powerful theory.
Henri Lebesgue (1902, doctoral thesis) developed a fundamentally different approach: instead of partitioning the domain (x-axis) into intervals, Lebesgue integration partitions the range (y-axis) into intervals and measures the "size" of the set of x-values mapping into each range interval, using measure theory. This handles far more functions than Riemann integration (including the Dirichlet function, which integrates to 0 under Lebesgue's definition), and provides the foundation for modern probability theory and functional analysis.
Integrals over infinite intervals, or of functions with singularities, are defined as limits: ∫₁^∞ (1/x²)dx = lim(b→∞) ∫₁ᵇ (1/x²)dx = lim(b→∞) [1 − 1/b] = 1. Such "improper integrals" require careful limit analysis — some converge to a finite value, others diverge to infinity.
Integration extends to functions of several variables: double integrals compute volumes under surfaces, triple integrals compute higher-dimensional analogues. Stokes' theorem (a vast generalisation encompassing the Fundamental Theorem of Calculus, Green's theorem, and the divergence theorem as special cases) relates the integral of a differential form over the boundary of a manifold to the integral of its derivative over the manifold's interior — a cornerstone result of differential geometry with deep applications in electromagnetism (Maxwell's equations) and fluid dynamics.
Leibniz introduced the integral sign ∫ (an elongated "S," for "summa," Latin for sum) in 1675. This notation, along with dy/dx for derivatives, proved more extensible and practical than Newton's fluxion notation, which is why Leibniz's symbols dominate calculus notation today even though the priority dispute over who invented calculus first was never fully resolved to either side's satisfaction during their lifetimes.