Integration

From definite integrals to the Lebesgue integral

About This Series

Integration is an operation that expresses accumulation, deeply connected to differentiation through the fundamental theorem of calculus. It is applied widely, from computing areas and volumes to probability theory and physics. This series starts from the indefinite and definite integrals of high-school mathematics and proceeds step by step through multivariable multiple integrals, line and surface integrals, and on to the Lebesgue integral.

Integration is a central concept of analysis and an indispensable tool across every field, including physics, engineering, economics, and probability theory.

Learn by Level

Introductory

High-school mathematics

  • Indefinite integrals
  • Definite integrals and area
  • Substitution and integration by parts
  • Applications of integration

Basic

University years 1–2

  • The Riemann integral
  • Improper integrals
  • Multiple integrals
  • Change of variables
  • Iterated integrals

Intermediate

University years 3–4

  • Line integrals
  • Surface integrals
  • Green's and Stokes' theorems
  • The divergence theorem

Advanced

Graduate level

  • Lebesgue measure
  • The Lebesgue integral
  • Convergence theorems
  • $L^p$ spaces

Learning Path

Intro High school Basic Univ. Y1–2 Interm. Univ. Y3–4 Advanced Graduate Intro: indefinite/definite integrals, substitution, by parts Basic: Riemann integral, multiple integrals, improper integrals Interm.: line integrals, surface integrals, vector calculus Adv.: Lebesgue measure, convergence theorems, Lp spaces Geometric meaning of integration Definite integral (area) D Multiple integral (region) Line integral (path)
Figure 1. The learning path and the geometric meaning of integration: the definite integral represents area, the multiple integral represents integration over a region, and the line integral represents integration along a path.

Main Topics

Definite Integrals and Area

The basic techniques for computing the area of a region bounded by a curve and an axis.

Multiple Integrals

Integration of multivariable functions, and the computation of volume and mass.

Line and Surface Integrals

Integration of vector fields along curves and over surfaces.

The Lebesgue Integral

The modern theory of integration based on measure theory.

Frequently Asked Questions

What is integration?

Integration is an operation that expresses accumulation, deeply connected to differentiation through the fundamental theorem of calculus. Geometrically it can be understood as the area under a curve or the volume of a solid, but more generally it represents the accumulation of a quantity such as mass or probability. The indefinite integral finds an antiderivative, while the definite integral computes the accumulated quantity over an interval as a number.

What is the difference between the Riemann integral and the Lebesgue integral?

The Riemann integral partitions the domain (the $x$-axis) to approximate area, whereas the Lebesgue integral partitions the range (the $y$-axis) and weights by measure. The Lebesgue integral applies to a wider class of functions, and its convergence theorems (the monotone convergence theorem, the dominated convergence theorem) are powerful. It is the standard integral in probability theory and functional analysis.

How do line integrals and surface integrals differ?

A line integral integrates a scalar or vector field along a curve (a one-dimensional path) and is used, for example, to compute work. A surface integral integrates over a surface (two-dimensional) and is used to compute flux. The two are related through Green's theorem and Stokes' theorem.

Why is a change of variables needed in multiple integrals?

Depending on the shape of the region of integration, computing in Cartesian coordinates can be difficult. For a circular region, switching to polar coordinates, or for a spherical region, to spherical coordinates, simplifies the integral greatly. A change of variables requires multiplying by the Jacobian (the Jacobian determinant).

In what fields is integration applied?

Integration is an indispensable tool in nearly every field of science and technology, including physics (motion, energy, electromagnetism), engineering (signal processing, control theory), economics (consumer surplus, expected value), probability theory (probability density functions, expectation), and statistics (distribution functions).

References