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

Lecture

Multiple Integration: Part 2

Description

This lecture covers the concept of double integrals over unbounded domains in R^2, introducing nested open bounded domains and the decomposition of continuous functions. It also explores the convergence of increasing and bounded sequences of double integrals, providing examples with exponential functions.

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Multiple integral

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

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 .

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Improper integral

In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions.

Nowhere continuous function

In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some such that for every we can find a point such that and . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.

Related lectures (31)

Generalized Integrals: Bounded Interval

Introduces generalized integrals on a bounded interval, discussing convergence, divergence, comparison criteria, variable substitution, and corollaries.

Generalized Integrals: Definitions and Criteria

Covers the definition of generalized integrals and comparison theorems for convergence.

Generalized Integrals: Definition and Convergence

Covers the definition and convergence of generalized integrals over bounded and unbounded intervals.

Multiple Integration: Fubini Theorem

Explores multiple integration in R², focusing on double integrals over closed rectangles and the Fubini theorem.

Generalized Integrals: Convergence and Divergence