double integral

(noun)

An integral extended to functions of more than one real variable

Related Terms

Examples of double integral in the following topics:

  • Double Integrals Over Rectangles

    • Double integrals over rectangular regions are straightforward to compute in many cases.
    • Formulating the double integral , we first evaluate the inner integral with respect to $x$:
    • We could have computed the double integral starting from the integration over $y$.
    • Double integral as volume under a surface $z = x^2 − y^2$.
    • Use double integrals to find the volume of rectangular regions in the xy-plane

  • Double Integrals Over General Regions

    • Double integrals can be evaluated over the integral domain of any general shape.
    • We studied how double integrals can be evaluated over a rectangular region.
    • The integral domain can be of any general shape.
    • In this atom, we will study how to formulate such an integral.
    • Double integral over the normal region $D$ shown in the example.

  • Green's Theorem

    • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
    • Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
    • In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.
    • In plane geometry and area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
    • Green's theorem can be used to compute area by line integral.

  • Triple Integrals

    • For $T \subseteq R^3$, the triple integral over $T$ is written as
    • Notice that, by convention, the triple integral has three integral signs (and a double integral has two integral signs); this is a notational convention which is convenient when computing a multiple integral as an iterated integral.
    • We have seen that double integrals can be evaluated over regions with a general shape.
    • The extension of those formulae to triple integrals should be apparent.
    • By calculating the double integral of the function $f(x, y) = 5$ over the region $D$ in the $xy$-plane which is the base of the parallelepiped: $\iint_D 5 \ dx\, dy$

  • Volumes

    • A volume integral is a triple integral of the constant function $1$, which gives the volume of the region $D$.
    • Using the triple integral given above, the volume is equal to:
    • For example, if a rectangular base of such a cuboid is given via the $xy$ inequalities $3 \leq x \leq 7$, $4 \leq y \leq 10$, our above double integral now reads:
    • Triple integral of a constant function $1$ over the shaded region gives the volume.
    • Calculate the volume of a shape by using the triple integral of the constant function 1

  • Parametric Surfaces and Surface Integrals

    • A surface integral is a definite integral taken over a surface .
    • It can be thought of as the double integral analog of the line integral.
    • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).
    • Surface integrals have many applications in physics, particularly within the classical theory of electromagnetism.
    • We will study surface integral of vector fields and related theorems in the following atoms.

  • Productivity: Argyris

    • Argyris's theory of single- and double-loop learning has been applied to management theory to suggest the best ways for employees to learn.
    • In double-loop learning, the entities question the values, assumptions, and policies that led to the actions in the first place; if they are able to view and modify those values, then second-order or double-loop learning has taken place.
    • This is a more integrative, process-oriented, and collaborative approach.
    • Double-loop learning may lead to a change in the original strategy or goals that the company had in the first place.
    • Identify Chris Argyris's key contributions to organizational theory through single-loop and double-loop learning

  • A short history of accounting and double entry bookkeeping

    • In my opinion, Goethe was exaggerating when he called double entry bookkeeping "one of the most beautiful discoveries of the human spirit".
    • At any rate, the discovery of double-entry bookkeeping was undeniably important, because, as Wikipedia explains:
    • Double-Entry Bookkeeping is a system that ensures the integrity of the financial values recorded in a financial accounting system.
    • Here is a simple example to give you a feel for the way that double entry bookkeeping works:
    • First, a single transaction affects two accounts (a double-entry).

  • Hybridization in Molecules Containing Double and Triple Bonds

    • Hybridized orbitals are very useful in explaining of the shape of molecular orbitals for molecules, and are an integral part of valence bond theory.
    • Ethene (C2H4) has a double bond between the carbons.
    • In this case, sp hybridization leads to two double bonds.
    • The remaining, non-hybridized p-orbitals overlap for the double and triple pi bonds.
    • Describe the role of hybridization in the formation of double and triple bonds.

  • Double-Stranded RNA Viruses: Retroviruses

    • Retroviruses are viruses that are able to reverse transcribe their RNA genome into DNA, which is then integrated into a host genome.
    • A special variant of retroviruses are endogenous retroviruses, which are integrated into the genome of the host and inherited across generations.
    • Once in the host's cell, the RNA strands undergo reverse transcription in the cytoplasm and are integrated into the host's genome, at which point the retroviral DNA is referred to as a provirus.
    • Their RNA is reverse-transcribed into DNA, which is integrated into the host cell's genome (when it becomes a provirus), and then undergoes the usual transcription and translation processes to express the genes carried by the virus .
    • This diagram depicts the viral life cycle of HIV, from infection, integration into a host genome, reconstruction, and formation of new viral particles.