integral

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The definite integral ∫ₐᵇ f(x)dx gives a number — the area of a curvilinear trapezoid bounded by the graph of f(x), the x-axis, and verticals x=a and x=b. It is defined as the limit of Riemann sums: Σ f(ξᵢ)Δxᵢ as the maximum length of partition intervals tends to zero. The indefinite integral is a family of antiderivatives, i.e., functions F such that F'=f. The connection is given by the Newton-Leibniz formula: ∫ₐᵇ f = F(b)−F(a).

History

Archimedes calculated the area of a parabolic segment by the method of exhaustion, close to integration. In the 17th century, Isaac Newton and Gottfried Leibniz independently reduced integration to finding an antiderivative — this is the fundamental theorem of calculus. Later, Bernhard Riemann gave a rigorous definition through limits of sums.

How it works

It’s like tiling a shaped area with mosaic: tiny tiles almost exactly cover the figure, and their total area is the desired quantity. Mathematically, it’s the limit of sums of areas of rectangles as their number increases without bound.

💡 The integral sign ∫ is a stylized letter ſ (long s), from Latin summa, proposed by Leibniz. Unlike the derivative, the integral smooths a function: even a highly jagged curve can have a smooth integral function.

Let f: [a,b] → ℝ be bounded. A partition: a = x₀ < x₁ < ... < xₙ = b. Riemann sum: Σ_{i=1,...,n} f(ξᵢ)(xᵢ−xᵢ₋₁), where ξᵢ ∈ [xᵢ₋₁, xᵢ]. The Riemann integral is the limit of such sums as maxᵢ (xᵢ−xᵢ₋₁) → 0, if the limit exists and does not depend on the choice of ξᵢ. Functions with a finite number of discontinuities are Riemann integrable. The Lebesgue integral generalizes to a broader class of functions via measure. Properties: linearity, additivity over intervals. Applied in physics to compute work, mass, flux; in probability theory to find probabilities via densities.

History

Eudoxus (c. 370 BC) and Archimedes (287–212 BC) used the method of exhaustion, the ancient analog of integration. In the 17th century, independent works of Newton and Leibniz formalized the integral as the inverse of the differential. Riemann in 1854 gave the first formal definition, and in the early 20th century, Henri Lebesgue constructed an integral based on measure, solving fundamental problems of interchange of limits and integral.

How it works

The fundamental theorem of calculus: if f is continuous on [a,b] and F is any antiderivative, then ∫ₐᵇ f(x)dx = F(b)−F(a). Computation reduces to finding an antiderivative. Techniques: substitution, integration by parts (derivative of a product), partial fractions. For complex integrands, numerical methods (trapezoidal, Simpson's) or Fourier/Laplace transforms are used.

\int_{a}^{b} f(x) dx = F(b) - F(a)

∫ₐᵇ f(x) dx — definite integral of f over [a,b]; F — any antiderivative of f (F' = f); a, b — limits of integration.

\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

n — number of subdivision intervals tending to infinity; Δx = (b−a)/n — length of each subinterval; xᵢ* — arbitrary point in the i-th subinterval; Σ — sum over i from 1 to n.

Open problems & fun facts

Creation of a universal algorithm for symbolic integration of all elementary functions (Risch problem); Exact computation of high-dimensional multiple integrals in quantum physics and financial mathematics problems

💡 The Poisson integral ∫_{-∞}^{∞} e^{-x²} dx = √π, or Gaussian integral, appears unexpectedly in probability theory and statistics, and its square is easily computed by switching to polar coordinates.

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integral

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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