Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Fundamental Theorem Statement

Description

This lecture covers the statement of the fundamental theorem of integral calculus, which states that for a continuous function f on a closed interval [a, b], the function G defined as the integral of f is a primitive of f. It also explains that if F is another primitive of f, then the definite integral of f over [a, b] is equal to F(b) minus F(a). The lecture emphasizes the importance of the mean value theorem in extending the functions G and F by continuity. Additionally, it discusses the properties and implications of the theorem.

In MOOCs (2)

Analysis I

Analysis I (part 7) : Indefinite and definite integrals, integration (selected chapters)

Official source

Quiz

Related concepts (23)

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.

Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves.

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .

Itô calculus

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes: where H is a locally square-integrable process adapted to the filtration generated by X , which is a Brownian motion or, more generally, a semimartingale.

Related lectures (36)

Integral Calculus: Fundamentals and Applications

Explores integral calculus fundamentals, including antiderivatives, Riemann sums, and integrability criteria.

Integral Calculus: Definite and Indefinite Concepts

Covers basic tools for definite integrals, antiderivatives, and integral calculus applications.

Integral: change of variable formula

Explores the change of variable formula for integrals and its applications.

Fundamental Theorem of Integral Calculus

Explores the fundamental theorem of integral calculus, area calculation, derivatives, and proof techniques.

Integral Calculus: Darboux Sums