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Related Concept Videos

Calculation of Volume of Solids by Integration01:27

Calculation of Volume of Solids by Integration

Volume calculation often begins with simple geometric solids. For example, the volume of a rectangular box is obtained by multiplying the area of its base by its height. This straightforward approach relies on the fact that the cross-sectional area of the box remains constant throughout its length. Many real-world objects, however, do not have uniform cross-sections, and their volumes cannot be determined using elementary geometric formulas.To address this limitation, the Slicing Method...
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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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Estimating the distance traveled by a vehicle using its recorded velocity over time is a common problem in physics and engineering. When velocity data is available at discrete time intervals, rather than as a continuous function, numerical integration methods such as the trapezoidal rule are often employed to approximate the total displacement.The trapezoidal rule works by dividing the total time interval into several equal segments. Within each segment, the recorded velocities at the endpoints...
Triple Integrals over General Regions01:28

Triple Integrals over General Regions

Triple integrals over general bounded regions extend the concept of double integrals from planar domains to three-dimensional solids. A solid region E in space is commonly enclosed within a rectangular box B, and a continuous function f(x, y, z) is integrated over the region by defining F such that it coincides with f on E and is zero outside the solid. The triple integral is therefore expressed as\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}The existence of the integral requires that f...
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Changing the order of integration can make a triple integral easier to evaluate without changing the solid region being measured. In this example, the solid is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The goal is to integrate ex over this three-dimensional region, so the main task is to describe the boundaries in an order that leads to the simplest calculation.One possible setup uses x as the innermost variable. In this arrangement, each line...
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For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...

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Updated: Jun 10, 2026

Rapid Setup of Tissue Microarray and Tiled Area Imaging on the Multiplexed Ion Beam Imaging Microscope Using the Tile/SED/Array Interface
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A fast and accurate algorithm for QTAIM integration in solids.

A Otero-de-la-Roza1, Víctor Luaña

  • 1Departamento de Química Física y Analítica, Facultad de Química, Universidad de Oviedo, 33006 Oviedo, Spain. alberto@carbono.quimica.uniovi.es

Journal of Computational Chemistry
|July 21, 2010
PubMed
Summary
This summary is machine-generated.

A novel algorithm, QTREE, rapidly calculates atomic properties in crystals using recursive subdivision. This method significantly speeds up computations for solid-state densities compared to traditional approaches.

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Area of Science:

  • Computational chemistry
  • Solid-state physics
  • Quantum chemistry

Background:

  • Calculating atomic properties in solids is crucial for understanding material behavior.
  • Traditional methods often struggle with the complexity of solid-state densities, leading to slow computation times.
  • The quantum theory of atoms in molecules provides a framework for defining atomic properties within a crystal.

Purpose of the Study:

  • To introduce a new, efficient algorithm named QTREE for calculating atomic properties in crystalline solids.
  • To demonstrate the speed and accuracy of QTREE compared to existing methods.
  • To enable the rapid computation of atomic properties for all atoms within a crystal.

Main Methods:

  • QTREE algorithm based on recursive subdivision of a symmetry-reduced Wigner-Seitz cell wedge.
  • Utilizes a union of tetrahedra and β-spheres for enhanced performance.
  • Applies to both analytical and interpolated electron densities.

Main Results:

  • QTREE achieves significant speedups, calculating properties in seconds to minutes.
  • Demonstrates convergence to accurate values with increasing precision.
  • Successfully determined basin volumes and charges for 11 test crystals.

Conclusions:

  • QTREE offers a substantial improvement in computational efficiency for atomic property calculations in solids.
  • The method is robust and accurate, providing reliable results for crystalline materials.
  • QTREE facilitates faster and more comprehensive analysis of atomic properties in solid-state systems.