Volume – wikipedia o gosh corpus christi

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance ( solid, liquid, gas, or plasma) or shape occupies or contains. [1] Volume is often quantified numerically using the SI derived unit, the cubic metre. 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. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape’s boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not additive. [2]

In the International System of Units (SI), the standard unit of volume is the cubic metre (m 3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = (10 cm) 3 = 1000 cubic centimetres = 0.001 cubic metres,

Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot. Related terms [ edit ]

Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, and to liquids, grain, or the like, which take the shape of that which holds them". [4] (The word capacity has other unrelated meanings, as in e.g. capacity management.) Capacity is not identical in meaning to volume, though closely related; the capacity of a container is always the volume in its interior. Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are closely related: one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the capacity of a vehicle’s fuel tank is rarely stated in cubic feet, for example, but in gallons (a gallon fills a volume of 0.1605 cu ft).

The density of an object is defined as the ratio of the mass to the volume. [5] The inverse of density is specific volume which is defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied.

Let the radius be r and the height be h (which is 2 r for the sphere), then the volume of cone is 1 3 π r 2 h = 1 3 π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 1 , {\displaystyle {\frac {1}{3}}\pi r^{2}h={\frac {1}{3}}\pi r^{2}\left(2r\right)=\left({\frac {2}{3}}\pi r^{3}\right)\times 1,}

Now ∫ − r r π r 2 d x − ∫ − r r π x 2 d x = π ( r 3 + r 3 ) − π 3 ( r 3 + r 3 ) = 2 π r 3 − 2 π r 3 3 . {\displaystyle \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}}.}

This formula can be derived more quickly using the formula for the sphere’s surface area, which is 4 π r 2 {\displaystyle 4\pi r^{2}} . The volume of the sphere consists of layers of infinitesimally thin spherical shells, and the sphere volume is equal to ∫ 0 r 4 π r 2 d r = 4 3 π r 3 . {\displaystyle \int _{0}^{r}4\pi r^{2}\,dr={\frac {4}{3}}\pi r^{3}.} Cone [ edit ]

However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally thin circular 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.