What is Volume: Definition and 1000 Discussions

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 3D shape occupies or contains. 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 complicated shapes can be calculated with 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 just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.
In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

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  1. ttpp1124

    Solving for the speed of volume

    Is this correct?
  2. Ishika_96_sparkles

    I Feynman's Lectures volume III (Ch:8) -- Resolution of vector states

    In the section 8-2 dealing with resolving the state vectors, we learn that |\phi \rangle =\sum_i C_i | i \rangle and the dual vector is defined as \langle \chi | =\sum_j D^*_j \langle j |Then, the an inner product is defined as \langle \chi | \phi \rangle =\sum_{ij} D^*_j C_i \langle j | i...
  3. A

    I Average of the B-field over a volume and surface integrals

    Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces. More specifically, he says in page...
  4. O

    I What is meant by rate of change with respect to volume?

    In physics we often come across $$\rho=\dfrac{dq}{dV}$$ Does it mean: ##(i)## ##\displaystyle \lim_{\Delta V \to 0} \dfrac{\Delta q}{\Delta V}## OR ##(ii)## ##\dfrac{\partial}{\partial z} \left( \dfrac{\partial}{\partial y} \left( \dfrac{\partial q}{\partial x} \right) \right)## What does...
  5. A

    Bernoulli's equation from an elemental fixed streamtube control volume

    Elemental fixed streamtube control volume from Professor White’s textbook “Fuid Mechanics”: I was unable to develop the intermediate steps for the following approximations: (continuity equation according to the book ) Where and (Momentum equation according to the book) In...
  6. CrosisBH

    Maximizing the volume of a cylinder

    Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it. The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for...
  7. G

    Determing the center of gravity of a shaded section

    Determine the volume of the shaded area around the Y-axis by using the theorem of Pappus Guldinus, where value of R = 143,3 cm. a) Determine the area of the shaded section. b) Determine the center of gravity of the shaded section. c) Detrmine the volume by using the theorem of Pappus Guldinus...
  8. rjomega

    Effect on Volume of a Change in the Pressure of Compressible Gas

    Will the available Volume of oxygen gas for use of patients increase when the pressure decreases from 12.4 MPa to 500 KPa? Is using boyle's law the right way to calculate the available volume?
  9. A

    MHB Find the volume of the hexagonal-shaped plastic box

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  10. Robert Friz

    I Area & Volume Naming Conventions: 4 Questions

    1. Area is the naming convention assigned to that which is within a closed diagram in the x-y dimensions. 2. Area is also the naming convention used in simplified Lorentzian diagrams in the x-t dimensions. 3. Volume is the naming convention used to that which is within a closed vessel in the...
  11. L

    A Volume element in Spherical Coordinates

    For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?
  12. A

    Thermodynamics energy balance for control volume

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  13. J

    Volume fraction of multiple phases

    Afternoon all, Hopefully somebody can help me, I'm doing my final year project and it's looking at the effect of heat treatment on in17, when I run an XRD scan I found that I all the phases sort of hid behind the matrix and so can't really make them out. So I've been looking at using the SEM...
  14. archaic

    Volume of an oblique circular cone

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  15. F

    I Demonstration of comoving volume between 2 redshifts

    1) I can't manage to find/justify the relation ##(1)## below, from the common relation ##(2)## of a volume. 2) It seems the variable ##r## is actually the comoving distance and not comoving coordinates (with scale factor ##R(t)## between both). The comoving volume of a region covering a solid...
  16. M

    Calculating the work needed to compress a volume of air

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  17. A

    Chemistry Constant Volume Heat of Combustion from heat capacity of calorimeter

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  18. brotherbobby

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  19. karush

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    ?
  20. jisbon

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  21. A

    Volume average of mass function

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  22. A

    Control volume and the momentum theorem

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  23. L

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  24. D

    Chemistry Adjust the volume to maximize the product yield

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  25. M

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  26. Aletag

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  27. B

    A The derivation of the volume form in Ricci tensor

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  28. D

    Isothermal process involving changes in Volume and Pressure

    P1 = 2 bar V1 = 5.1L P2= 1bar V2 = V1P1/P2 = 10,2L, so the volume of gas would double? or should the absolute pressure be taken into account P1= 2bar (3bar absolute), V1=5.1L P2= 1 bar V2 = 15,3L?
  29. archaic

    Volume of a Frustum - Get Help Now

    EDIT: I thought I was in the math section for homework, sorry! My work is wrong, I don't see why though. Help much appreciated :)
  30. V

    Finding volume using integration

    I know that the formula for volume is equal to the definite integral ∫A(x)dx, where A(x) is the cross sectional. I found the definite integral where b=5 and a=0, for ∫4x2dx. I obtained the answer 500/3, however this was incorrect, and I'm unsure of where I went wrong? Thank you.
  31. C

    Phase space volume with a potential (microcanonical ensemble)

    I don't know how to solve that integral, and to calculate the number of microstates first, then aply convolution and then integrate to find the volume of the phase space seems to be more complicated. Any clue on how to solve this? Thank you very much.
  32. M

    Find the volume of the solid formed by the rotation around the y=0

    Hi, I find this... Please tell me your opinion on this. Thanks.
  33. N

    MHB Decide volume given two functions

    Sorry if i made any language errors, english is not my first language. Question: An area in the first quadrant (x=>0,y=>0) is limited by the axis and the graphs to the functions f(x)=x^2-2 and g(x)=2+x^2/4. When the area rotates around the y-axis a solid is created. Calculate the volume of...
  34. N

    MHB Calculate volume of a solid rotating around the y-axis

    Sorry if i made any language errors, English isn't my first language. Question: The limited area in the plane is created when the space between the line y=1 and the graph to the function f(x)=3*x/(x^2+1) rotates around the y-axis. Calculate the volume of the solid.I want to sum up all the...
  35. D

    I 4-dimensional element volume

    I've come across discussions about the invariant properties of the 4 volume dV=dxdydzdt, but have yet to see its use in many equations. What is this object mostly used in and how is it or would it be used in quantum physics, cosmology, and relativity?
  36. C

    Change in a marine mammal's lung volume when diving to 200 meters

    I attempt the solution on the attachment. The answer is 0.344 litre. Do I change 7L to m, so it is 0.007 cubic meters
  37. Sabra_a

    Mass fraction and volume of a gas in a cylinder

    I have attached the full answer in PDF file. I'm not sure about the answers. will really appreciate if they get checked
  38. B

    Chemistry Computation of Liquid/Vapor Result during a Volume Expansion

    I am searching for the appropriate methodology/equation(s) to step beyond Boyle's Law to account for the phase change and solve this problem. All suggestions/guidance is greatly appreciated! Bruce
  39. NP04

    Derivative of a cylinder's volume

    Using product rule, we have: [d/dx] (πr^2)(h) = (πr^2)(1 ) + (2πr)(h) Why is the two there? V = 2 πrh+2πr^2 The derivative of h is 1, not 2. Please help!
  40. Diracobama2181

    A Volume Element for Isotropic Harmonic oscillator

    I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator. I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$. What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...
  41. K

    Gibbs' theorem and partial molar volume

    In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a...
  42. Philip Koeck

    A The probability of finding R out of N bosons in one half of a volume

    For the probability of finding R out of N (indistinguishable) bosons in one half of a volume with a total of 2g states (g in each half) I get the following expression: PR = WR / WT where WT is the number of ways of distributing N particles in the total volume: WT = (N+2g-1)! / (N! (2g-1)!)...
  43. A

    Volume of ice needed to mitigate ocean warming since 1871

    According to following study 436 x 10^21 J of energy have been absorbed by the Earth's oceans since 1871. https://www.pnas.org/content/116/4/1126 What thickness of ice covering the globe would be needed to melt in order to absorb this amount of energy, assuming that all energy goes towards the...
  44. jisbon

    Calculating the volume of the solid in this graph

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  45. falcon999

    Effect on air pressure and volume in an enclosed container

    Summary: I wish to understand if bubble formation in milk while being sloshed around, or the formation of separation layers will affect both the pressure and volume of the air head space above the milk Hi all : ) I have a basic physics question and sorry if its a very silly question: Let's...
  46. Beelzedad

    I Swapping the order of surface and volume integrals

  47. K

    Help With MCNP Volume definition

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  48. Beelzedad

    I Is interchanging the order of the surface and volume integrals valid here?

    Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. Consider the following multiple integral: ##\displaystyle A=\iiint_{V'} \left[ \iint_S \dfrac{\cos(\hat{R},\hat{n})}{R^2} dS \right] \rho'\ dV' =4 \pi\ m_s## where...
  49. Izazo

    Outlet Volume flow rate from a steam turbine

    I am working Organic Rankine Cycle. I studied a number of research papers and in most of them, they have calculated outlet volume flow rate from the turbine or expander, but have not mentioned the calculations. So here are the available data; Pin = Turbine Inlet Pressure, 2.5 MPa Pout = Turbine...
  50. lfdahl

    MHB Volume of the solid obtained by rotating R around the line y=x

    Let $R$ be the region $\left\{(x, y) : 0 \leq x \leq 1, 3^x − x − 1 \leq y \leq x\right\}$. Find the volume of the solid obtained by rotating $R$ around the line $y = x$.
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