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. _N3WTON_

    What is the Volume of a Region Bounded by Double Integrals with a Max Function?

    Homework Statement Evaluate: \int_{0}^{a} \int_{0}^{b} e^{max(b^{2}x^{2}, a^{2}y^{2})} dy \hspace{1 mm} dx Where a and b are positive. Homework EquationsThe Attempt at a Solution I'm having some trouble getting started with this problem, mostly because I am not very familiar with the...
  2. L

    Finding volume of a nose cone with a given r with integration

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  3. O

    Finding remaining volume of water in a cylinder.

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  4. S

    Changes in volume of a liquid affects?

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  5. C

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  6. B

    Volume of solid x^2 + (y-1)^2 =1 about y-axis

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  7. MathematicalPhysicist

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  8. gfd43tg

    How do gas reactions affect volume in a fixed container?

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  9. G

    Volume Fluxes and Associated Velocities

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  10. G

    Help: Volume Fluxes and Associated Velocities

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  11. H

    Enthelpy of Reaction under Constant Volume?

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  12. G

    MHB Calculating Volume of Ellipsoidal Propane Tanks

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  13. C

    Minimum volume of phase space

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  14. E

    Pressure Calculations through Volume Differences

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  15. P

    "best" value for volume of a box

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  16. gfd43tg

    How Do Levenspiel Plots Determine CSTR Volume for Increasing Reaction Rates?

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  17. J

    Finding Mass Through Percent by Volume

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  18. W

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  19. RJLiberator

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  20. M

    Using shell method to find the volume of a solid

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  21. C

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  22. T

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  23. M

    Volume Change in Phase Transitions

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  24. ZetaOfThree

    Volume of Brillouin zone is the same as Fourier primitive parallelepip

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  25. squelch

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  26. I

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  27. N

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  28. O

    Volume bounded by cylinder and planes

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  29. V

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  30. wolf1728

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  31. G

    How to calculate that shape inside volume?

    Dear everybody here, please i need help at that attached image of round shape as shown, my questions is what is the formula which used for calculating the volume of that shape?, the cylinder of D1, and D2, , easily can be calculated by (Pi/4*D2*L), so how about in the middle? will...
  32. STEMucator

    Solving a Tough Integral: Volume of First Octant Region

    Homework Statement I ran into this tough integral. I am asked to compute the volume of the following region using a double integral with ##a, b, c > 0##. The first octant region bounded by the co-ordinate planes ##x=0##, ##y=0##, ##z=0## and the cylinders ##a^2y = b(a^2-x^2)## and ##a^2z =...
  33. B

    Understanding Cell Volume: Why It's Independent of Choice

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  34. karush

    MHB Find the volume by revolving around the x-axis without expanding

    A solid is generated when the region in the first quadrant enclosed by the graph of , y=(x^2+1)^2 The line x=1 , the x-axis, and the y-axis is revolved about the x-axis. Find the volumn $$ V=\int_{a}^{b}\pi\left[f(x)\right]^2 dx =\int_{0}^{1}\pi\left[\left(x^2+1\right)^2\right]^2 \,dx...
  35. J

    Pharmacology: Clearance vs Volume of Distribution

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  36. N

    What is the Correct Equation for Volume and Its Units in this Physics Problem?

    Homework Statement In the equation dm = δ x 2∏rLdr Where δ = density, and 2∏rL = volume How is it that the volume can be 2∏rL? The units of r is (metres) and the units of L is (metres) which leads to m2 (Area) Should it not be ∏r2L for volume? The Attempt at a Solution...
  37. I

    MHB How to find the volume of a parallelepiped using determinants?

    find the parallelepiped determined by the vector $a= \left\langle 1,2,3 \right\rangle$, $b= \left\langle -1,1,2 \right\rangle$, $c= \left\langle 2,1,4 \right\rangle$ find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. $P(-2,1,0)$, $Q(2,3,2)$, $R(1,4,-1)$, $S(3,6,1)$ 1...
  38. G

    Temperatue and Presssure after volume change

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  39. R

    Understanding Volume of Oblique Cylinders through Integration

    Hi :) I'm doing my A-Levels and have a maths investigation project for which I decided to model the working of a shishi-odoshi. (http://en.wikipedia.org/wiki/Shishi-odoshi) The shape of the water in the shishi-odoshi is a cylindrical segment and I want to use integration to find the...
  40. phosgene

    Maximise the volume of a rectangular prism with 2 constraints

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  41. Dethrone

    MHB Solids of Revolution - Negative Volume

    I encountered a problem where the answer I got was negative. Calculate the volume bounded by $y=x^2-5x+6$, $y=0$, about y-axis. An easy question that is best done with the cylindrical shell method: $$V=2\pi \int_{2}^{3} x(x^2-5x+6)\,dx$$ $$V=\frac{-5\pi}{6}$$ I think I know why it's...
  42. B

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    Folks, I am having difficulty correctly representing a mechanical system within a correct "control volume at an instant" in order to identify the various energy balance terms given below ##\displaystyle \dot E_{st}=\frac{d E_{st}}{dt}=\dot E_{in} - \dot E_{out}+ \dot E_g## (1) that...
  43. N

    Determine the mass of an object given the volume it has displaced

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  44. T

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  45. Dethrone

    MHB Volume of Region Bounded by x^2-y^2=16, y=0, x=8, about y-axis

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  46. S

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  47. J

    Volume of Region Bounded by x^2+2y^2=2, x+y+2z=2

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  48. J

    Compute volume below z=x^2+y and above [0,1]x[1,2]

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  49. E

    Deriving a volume formula, answer seems to be correct but is negative.

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  50. J

    Show rectangular box of given volume has minimum surface area when

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