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Volumes of Solids of Revolution

Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region defined by a curve in a two-dimensional plane. When this region is rotated about a fixed line, known as the axis of revolution, it generates a three-dimensional object with rotational symmetry. Such objects frequently arise in mathematical modeling, physics, and engineering applications.When the region being rotated lies directly against the axis...

Triple Integrals in Spherical Coordinates

Triple Integrals in Spherical Coordinates

Triple integrals in spherical coordinates provide an efficient method for evaluating volumes over regions with central symmetry, such as spheres. Instead of describing points by rectangular coordinates, spherical coordinates use three variables: 𝜌, 𝜃, and 𝜑. Here, 𝜌 is the distance from the origin, 𝜃 is the angle in the xy-plane measured from the positive x-axis, and 𝜑 is the angle measured downward from the positive z-axis.To derive the volume of a sphere, the solid region can be divided...

Calculation of Volume of Solids by Integration

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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...

Finding Volume Using Cross-Sectional Area

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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...

The Midpoint Rule for Double Integrals

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The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective...

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Multiple integrals provide a powerful mathematical framework for calculating physical quantities distributed throughout two- and three-dimensional regions. One important application is the determination of volume in objects with curved geometries, such as storage tanks, pipes, and reservoirs. Cylindrical coordinates are especially useful for systems with rotational symmetry because they simplify the description of circular and paraboloid-shaped regions.Consider a paraboloid-shaped water tank...

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Logically rectangular finite volume methods with adaptive refinement on the sphere.

Marsha J Berger¹, Donna A Calhoun, Christiane Helzel

¹Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|October 21, 2009

Summary

This summary is machine-generated.

New finite volume grids offer uniform cell size for spherical problems, improving computational efficiency. Adaptive mesh refinement with GeoClaw and well-balanced methods maintain equilibrium solutions for geophysical flows.

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Area of Science:

Computational fluid dynamics
Geophysical fluid dynamics
Numerical analysis

Background:

Traditional grids face limitations with spherical geometries and complex bathymetry.
Severe Courant number restrictions hinder computational efficiency in solving PDEs on spheres.

Purpose of the Study:

Introduce logically rectangular finite volume grids for 2D/3D spherical problems.
Demonstrate adaptive mesh refinement and well-balanced methods for geophysical flows.

Main Methods:

Developed novel finite volume grids with nearly uniform cell size.
Utilized GeoClaw software for adaptive mesh refinement.
Implemented well-balanced methods to preserve equilibrium solutions.

Main Results:

The new grids avoid severe Courant number restrictions.
Adaptive mesh refinement enhances solution accuracy and efficiency.
Well-balanced methods successfully maintain equilibrium states for shallow water equations over arbitrary bathymetry.

Conclusions:

Logically rectangular finite volume grids are effective for spherical PDEs.
GeoClaw and well-balanced methods provide robust solutions for geophysical fluid dynamics.
The approach enhances the simulation of phenomena like oceans at rest.