integration

Riemann Integral

A method of integration that defines the integral of a bounded function over an interval [a,b] based on the limit of Riemann sums as the subdivision steps approach zero.

Subdivision

A finite family of real numbers that divides the closed interval [a,b] into smaller intervals, with specific conditions on the endpoints.

Riemann Sum

The sum of areas of rectangles formed by a function and a subdivision, representing an approximation of the integral.

Equidistributed Subdivision

A subdivision where the intervals are of equal length, derived from an arithmetic progression.

Integrable Function

A bounded function f on [a,b] for which the limit of Riemann sums converges to a real number l as the subdivision steps approach zero.

Dummy Variable

The variable used in the integral notation, which does not affect the value of the integral.

Piecewise Continuous Function

A function that is continuous on subintervals of [a,b] and has finite limits at the endpoints of those intervals.

Continuous Function

A function that does not have any breaks, jumps, or discontinuities on the interval [a,b].

Integration by Parts

A technique for integrating products of functions, based on the product rule of differentiation.

Change of Variable

A method used in integration to simplify the integral by substituting a new variable for the original variable.

Rational Fraction

A fraction where the numerator and denominator are polynomials, which can be integrated using specific techniques based on their degrees.

Wallis Integral

A specific integral related to the Wallis product, often involving trigonometric functions.

Abelian Integrals

Integrals of a specific form that can be computed using particular substitutions or transformations.

Bioche’s Rule

A method for determining appropriate variable changes in integrals based on symmetry properties of the integrand.

Primitive Function

A function whose derivative is equal to the original function, often used in the context of finding integrals.