volume

Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume.

Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume. The metric units of volume uses metric prefixes, strictly in powers of ten. When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm 3 = 2.3 (cm) 3 = 2.3 (0.01 m) 3 = 0.0000023 m 3 (five zeros). Commonly used prefixes for cubed length units are the cubic millimetre (mm 3 ), cubic centimetre (cm 3 ), cubic decimetre (dm 3 ), cubic metre (m 3 ) and the cubic kilometre (km 3 ). The conversion between the prefix units are as follows: 1000 mm 3 = 1 cm 3 , 1000 cm 3 = 1 dm 3 , and 1000 dm 3 = 1 m 3 . The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm 3 = 1000 cm 3 = 0.001 m 3 . For the litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Various other imperial or U.S. customary units of volume are also in use, including: cubic inch, cubic foot, cubic yard, acre-foot, cubic mile; minim, drachm, fluid ounce, pint; teaspoon, tablespoon; gill, quart, gallon, barrel; cord, peck, bushel, hogshead. The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution, by rotating a plane curve around a line on the same plane. The washer or disc integration method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = \pi \int_a^b \left| f(x)^2 - g(x)^2\right|\,dx where f(x) and g(x) are the plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2\pi \int_a^b x |f(x) - g(x)|\, dx The volume of a region D in three-dimensional space is given by the triple or volume integral of the constant function f(x,y,z) = 1 over the region. It is usually written as: \iiint_D 1 \,dx\,dy\,dz. In cylindrical coordinates, the volume integral is \iiint_D r\,dr\,d\theta\,dz, In spherical coordinates (using the convention for angles with \theta as the azimuth and \varphi measured from the polar axis; see more on conventions), the volume integral is \iiint_D \rho^2 \sin\varphi \,d\rho \,d\theta\, d\varphi .

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