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గోళం, స్థూపం ల ఘనపరిమాణాల నిష్పత్తి '''2 : 3''' అని ఆర్కిమెడిస్ అనే శాస్త్రవేత్త కనుగొన్నాడు. <ref>{{cite web|url=http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title=Tomb of Archimedes: Sources|last=Rorres|first=Chris|publisher=Courant Institute of Mathematical Sciences|access-date=2007-01-02}}</ref>
=== Volume formula derivations ===
==== Sphere ====
The volume of a [[:en:Sphere|sphere]] is the [[:en:Integral|integral]] of an infinite number of infinitesimally small circular [[:en:Disk_(mathematics)|disks]] of thickness ''dx''. The calculation for the volume of a sphere with center 0 and radius ''r'' is as follows.
The surface area of the circular disk is <math>\pi r^2 </math>.
The radius of the circular disks, defined such that the x-axis cuts perpendicularly through them, is
: <math>y = \sqrt{r^2 - x^2}</math>
or
: <math>z = \sqrt{r^2 - x^2}</math>
where y or z can be taken to represent the radius of a disk at a particular x value.
Using y as the disk radius, the volume of the sphere can be calculated as
: <math> \int_{-r}^r \pi y^2 \,dx = \int_{-r}^r \pi\left(r^2 - x^2\right) \,dx.</math>
Now
: <math>\int_{-r}^r \pi r^2\,dx - \int_{-r}^r \pi x^2\,dx = \pi \left(r^3 + r^3\right) - \frac{\pi}{3}\left(r^3 + r^3\right) = 2\pi r^3 - \frac{2\pi r^3}{3}.</math>
Combining yields <math>V = \frac{4}{3}\pi r^3.</math>
This formula can be derived more quickly using the formula for the sphere's [[:en:Surface_area|surface area]], which is <math>4\pi r^2</math>. The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to
: <math> \int_0^r 4\pi r^2 \,dr = \frac{4}{3}\pi r^3.</math>
==== Cone ====
The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.
However, using calculus, the volume of a [[:en:Cone_(geometry)|cone]] is the [[:en:Integral|integral]] of an infinite number of infinitesimally thin circular [[:en:Disk_(mathematics)|disks]] of thickness ''dx''. The calculation for the volume of a cone of height ''h'', whose base is centered at (0, 0, 0) with radius ''r'', is as follows.
The radius of each circular disk is ''r'' if ''x'' = 0 and 0 if ''x'' = ''h'', and varying linearly in between—that is,
: <math>r \frac{h - x}{h}.</math>
The surface area of the circular disk is then
: <math> \pi \left(r\frac{h - x}{h}\right)^2 = \pi r^2\frac{(h - x)^2}{h^2}. </math>
The volume of the cone can then be calculated as
: <math> \int_0^h \pi r^2\frac{(h - x)^2}{h^2} dx, </math>
and after extraction of the constants
: <math>\frac{\pi r^2}{h^2} \int_0^h (h - x)^2 dx</math>
Integrating gives us
: <math>\frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right) = \frac{1}{3}\pi r^2 h.</math>
==== Polyhedron ====
== Volume in differential geometry ==
In [[:en:Differential_geometry|differential geometry]], a branch of [[:en:Mathematics|mathematics]], a '''volume form''' on a [[:en:Differentiable_manifold|differentiable manifold]] is a [[:en:Differential_form|differential form]] of top degree (i.e., whose degree is equal to the dimension of the manifold) that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a [[:en:Density_on_a_manifold|density]]. Integrating the volume form gives the volume of the manifold according to that form.
An [[:en:Orientation_(mathematics)|oriented]] [[:en:Pseudo-Riemannian_manifold|pseudo-Riemannian manifold]] has a natural volume form. In [[:en:Local_coordinates|local coordinates]], it can be expressed as
: <math>\omega = \sqrt{|g|} \, dx^1 \wedge \dots \wedge dx^n ,</math>
where the <math>dx^i</math> are [[:en:1-form|1-forms]] that form a positively oriented basis for the [[:en:Cotangent_bundle|cotangent bundle]] of the manifold, and <math>g</math> is the [[:en:Determinant|determinant]] of the matrix representation of the [[:en:Metric_tensor|metric tensor]] on the manifold in terms of the same basis.
== Volume in thermodynamics ==
In [[:en:Thermodynamics|thermodynamics]], the '''volume''' of a [[:en:Thermodynamic_system|system]] is an important [[:en:Extensive_parameter|extensive parameter]] for describing its [[:en:Thermodynamic_state|thermodynamic state]]. The '''specific volume''', an [[:en:Intensive_property|intensive property]], is the system's volume per unit of mass. Volume is a [[:en:Function_of_state|function of state]] and is interdependent with other thermodynamic properties such as [[:en:Pressure|pressure]] and [[:en:Thermodynamic_temperature|temperature]]. For example, volume is related to the [[:en:Pressure|pressure]] and [[:en:Thermodynamic_temperature|temperature]] of an [[:en:Ideal_gas|ideal gas]] by the [[:en:Ideal_gas_law|ideal gas law]].
== Volume computation ==
The task of numerically computing the volume of objects is studied in the field of [[:en:Computational_geometry|computational geometry]] in computer science, investigating efficient [[:en:Algorithm|algorithms]] to perform this computation, [[:en:Approximation_algorithm|approximately]] or [[:en:Exact_algorithm|exactly]], for various types of objects. For instance, the [[:en:Convex_volume_approximation|convex volume approximation]] technique shows how to approximate the volume of any [[:en:Convex_body|convex body]] using a [[:en:Oracle_machine|membership oracle]].
== మూలాలు ==
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