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Related Concept Videos

Theorems of Pappus and Guldinus: Problem Solving01:12

Theorems of Pappus and Guldinus: Problem Solving

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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Work Done During Volume Change01:17

Work Done During Volume Change

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In mechanics, work is done on an object when the force acting on it displaces the object. In thermodynamics, work done on a system can be estimated when the system's volume changes during any thermodynamic process.
Consider a gas confined to a cylinder fitted with a movable piston at one end. If the gas expands from volume V1 to volume V2, it exerts a force on the piston, such that the piston moves by a distance dr.
The work done by the gas on the piston can be expressed as
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Unsoundness of Aggregate due to Volume Change01:26

Unsoundness of Aggregate due to Volume Change

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Unsoundness in aggregates due to volume changes is primarily caused by the physical alterations aggregates undergo, such as freezing and thawing, thermal changes, and wetting and drying. Unsound aggregates, when subjected to these changes, result in volume change upon disintegration. This, in turn, contributes to the deterioration of concrete, including scaling, pop-outs, and cracking. Particular types of aggregates, such as porous flints, cherts, and those containing clay minerals, are...
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Uniform Depth Channel Flow: Problem Solving01:18

Uniform Depth Channel Flow: Problem Solving

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To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...
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Theorems of Pappus and Guldinus01:10

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The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
For finding the surface area, consider a differential line element that generates a ring with surface area dA when revolved.
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Line, Surface, and Volume Integrals01:15

Line, Surface, and Volume Integrals

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A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero...
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Related Experiment Video

Updated: Nov 7, 2025

A Volumetric Method for Quantification of Cerebral Vasospasm in a Murine Model of Subarachnoid Hemorrhage
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CT Volumetry of Convoluted Objects-A Simple Method Using Volume Averaging.

Rani Al-Senan1, Jeffrey H Newhouse2

  • 1Department of Radiology, Pennsylvania State University, Hershey, PA 17033, USA.

Tomography (Ann Arbor, Mich.)
|April 30, 2021
PubMed
Summary
This summary is machine-generated.

This study presents a simple, accurate method for measuring object volumes using computed tomography (CT). The technique is reliable for complex objects and does not require specialized software, making it useful for clinical and research applications.

Keywords:
CTpartial-volume averagingsegmentationvolumetry

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Area of Science:

  • Medical Imaging
  • Quantitative Analysis

Background:

  • Accurate object volume measurement using computed tomography (CT) is crucial but challenging, particularly for convoluted structures with blurred margins.
  • Volume averaging in CT can compromise the precision of traditional volumetry techniques.

Purpose of the Study:

  • To evaluate the accuracy of a straightforward method for object volumetry in CT scans.
  • To assess the method's efficacy on a phantom object designed to mimic challenging clinical scenarios.

Main Methods:

  • A phantom object composed of lucite beads in petroleum jelly was constructed and scanned using CT.
  • A simple region of interest (ROI) method was employed, utilizing object and surrounding material densities, and slice thickness for volume calculation.
  • The calculated volumes were compared against the phantom's true volume.

Main Results:

  • The developed method demonstrated high accuracy in determining object volumes.
  • Accuracy was maintained despite variations in slice thickness, image noise, reconstruction planes, spatial resolution, and ROI selection.

Conclusions:

  • This simple CT volumetry method is highly accurate and robust for complex objects.
  • The technique offers a practical, software-independent solution for accurate volume measurement in clinical and research settings.