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A cross-section view of a compression seal. In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections.
Calculus is the mathematical study ... who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections.
This file represents the Cavalieri's Principle in action: if you have the same set of cross sections that only differ by a horizontal translation, you will get the same volume. In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: [1]
The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be: ()
The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (d) between the axes. ′ = +
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus.
This is an application of Cavalieri's principle: volumes with equal-sized corresponding cross-sections are equal. Indeed, the area of the cross-section is the same as that of the corresponding cross-section of a sphere of radius h / 2 {\displaystyle h/2} , which has volume 4 3 π ( h 2 ) 3 = π h 3 6 . {\displaystyle {\frac {4}{3}}\pi \left ...
Any horizontal cross-section of a cloister vault is a square. This fact may be used to find the volume of the vault using Cavalieri's principle.Finding the volume in this way is often an exercise for first-year calculus students, [4] and was solved long ago by Archimedes in Greece, Zu Chongzhi in China, and Piero della Francesca in Renaissance Italy; [5] for more, see Steinmetz solid.
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