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

Lecture

Volume Calculation in R^3

In course

MATH-106(e): Analysis II

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

Description

This lecture covers the calculation of volumes of subsets in R^3 defined by inequalities, using double integrals over regular domains of type 1 and 2. The instructor demonstrates the application of the Fubini's theorem and provides examples to illustrate the concepts.

Instructor

Anna Lachowska

Official source

Related concepts (26)

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.

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.

Fubini's theorem

In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands.

Dirichlet integral

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: This integral is not absolutely convergent, meaning is not Lebesgue-integrable, because the Dirichlet integral is infinite in the sense of Lebesgue integration. It is, however, finite in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral.

Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution.

Related lectures (31)

Multiple Integration: Fubini Theorem

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

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Integrals in Higher Dimensions

Explores integrals in higher dimensions, emphasizing the versatility of integration methods and the significance of changing the order of integration.

Fubini Theorem: Double and Triple Integrals

Explores the Fubini theorem for double and triple integrals, focusing on regular domains and order of integration.

Analytic Continuation: Residue Theorem