Stewart, James (2008). Calculus: Early Transcendentals (6thed.). Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8.

Further information: History of calculus and Apothecaries' system Diagram showing how to measure volume using a graduated cylinder with fluid dram markings, 1926 In the Middle Ages, many units for measuring volume were made, such as the sester, amber, coomb, and seam. The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. [4] :73–74 In 1618, the London Pharmacopoeia (medicine compound catalog) adopted the Roman gallon [8] or congius [9] as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. [8] Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5mL (0.03–0.2USfloz; 0.04–0.2impfloz). [4] :8 a b "Volumes by Integration" (PDF). Rochester Institute of Technology. 22 September 2014. Archived (PDF) from the original on 2 February 2022 . Retrieved 12 August 2022.Benson, Tom (7 May 2021). "Gas Density". Glenn Research Center. Archived from the original on 2022-08-09 . Retrieved 2022-08-13. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m 3). The cubic metre is also a SI derived unit. [15] Therefore, volume has a unit dimension of L 3. [16] Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent cuboid resting on a common surface, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a( S) ≤ c ≤ a( T) for all such step regions S and T, then a( Q) = c. a b Le Système international d’unités[ The International System of Units] (PDF) (in French and English) (9thed.), International Bureau of Weights and Measures, 2019, ISBN 978-92-822-2272-0 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: 2.3cm 3 = 2.3 (cm) 3 = 2.3 (0.01 m) 3 = 0.0000023 m 3 (five zeros). [17] :143

a b c d e f Imhausen, Annette (2016). Mathematics in Ancient Egypt: A Contextual History. Princeton University Press. ISBN 978-1-4008-7430-9. OCLC 934433864. IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain" ". International Electrotechnical Vocabulary (in Japanese) . Retrieved 2023-09-19.Cardarelli, François (6 Dec 2012). Scientific Unit Conversion: A Practical Guide to Metrication (2nded.). London: Springer Science+Business Media. p.151. ISBN 978-1-4471-0805-4. OCLC 828776235. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). [2] [3]

Cengel, Yunus A.; Boles, Michael A. (2002). Thermodynamics: an engineering approach. Boston: McGraw-Hill. p.11. ISBN 0-07-238332-1. Mise en pratique for the definition of the metre in the SI" (PDF). International Bureau of Weights and Measures. Consultative Committee for Length. 20 May 2019. p.1. Archived (PDF) from the original on 13 August 2022 . Retrieved 13 August 2022. 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 fourth and higher dimensions, an analogous concept to the normal volume is the hypervolume.

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a b c d e f g h i j k l m n Treese, Steven A. (2018). History and Measurement of the Base and Derived Units. Cham, Switzerland: Springer Science+Business Media. ISBN 978-3-319-77577-7. LCCN 2018940415. OCLC 1036766223. Air displacement pipette is used in biology and biochemistry to measure volume of fluids at the microscopic scale. [14] Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories. There, volume of liquids is measured using graduated cylinders, pipettes and volumetric flasks. The largest of such calibrated containers are petroleum storage tanks, some can hold up to 1,000,000 bbl (160,000,000L) of fluids. [5] :399 Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. [5] :403