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. takgt7

    Calculating the volume in closed container?

    Hi, I have a hard time calculating volume in a closed container. Let's say, I have a flask with a volume 0.125ml, which is closed, and the pressure inside is 16.87kPa with temperature 294.7k and I did some reaction to create a gas inside the flask. The new pressure in the flask is 87.19kPa and...
  2. P

    Volume of ice cream cone triple integral

    Homework Statement Find the triple integral for the volume between a hemisphere centred at ##z=1## and cone with angle ##\alpha##.The Attempt at a Solution What I tried to do first was to get the radius of the hemisphere in terms of the angle ##\alpha##. In this case the radius is ##\tan...
  3. M

    Finding Volumes of Sphere & Circular Cone: Alpha from 0 to Pi

    Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the...
  4. F

    Volume moment to mass moment

    Hello, I was wondering if some one would be able to check my thought process on this. I have an object ( a boat) which I'm only able to calculate the tensors of the volume moment around its volume centroid. If i assume that the object has a uniform density ( which it does not, just bear with...
  5. physkim

    Volume integral of a function over tetrahedron

    Homework Statement Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$ over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$ Homework Equations I was able to solve it mathematically, but still can't figure out why the answer is so small. I only understand...
  6. N

    Floating on Ice: Calculating Volume Needed to Stay Afloat

    Homework Statement A person (with mass 60.0 kg) is located on a volume of ice, floating on the water. Calculate the smallest volume of the ice so that the person would remain above the water ( ice density = 917kg / m3)[/B]Homework Equations F= m xg Archimedes F = density x V xg [/B] The...
  7. abizan

    What is the largest possible volume for this cylinder?

    Homework Statement A rectangle with a perimeter of 40 cm is rotated around one of its sides creating a right cylinder. What is the largest possible volume for this cylinder? Homework Equations Volume of a cylinder= pi*r^2*h Perimeter of a rectangle= 2x + 2L The Attempt at a Solution I know...
  8. H

    Understanding G and γ in Volume Flow Rate Calculations

    Homework Statement what is G and γ mean here ? Homework EquationsThe Attempt at a Solution
  9. R

    How can a volume integral yield a vector field?

    I'm using the textbook Electricity and Magnetism by Purcell. In the section about continuous charge distributions I found the following formula \mathbf{E}(x,y,z)= \frac{1}{4\pi\epsilon_0 } \int \frac{\rho(x',y',z')\boldsymbol{\hat r} dx'dy'dz'}{r^{2}} . It's stated that (x,y,z) is fixed...
  10. A

    Calculating the Volume of a Rotated Solid Using Calculus

    Find the volume of the solid generated by rotating the region of the x-y plane between the line Y=4,the curve Y=3sin(x)+1 on the interval [-pi/2,3pi/2] about the line Y=4Hi I am having trouble setting up this problem my guess for the integral would be from -pi/2 to 3pi/2 of (4-3sinx+1)^2 because...
  11. G

    MHB Volume enclosed by rotating a curve segment

    The volume enclosed by rotating the segment of the curve $y = \frac{1}{2}|x-1|$ between $x = 0$ and $x = 2$ about the $x$-axis is equal to: Is it this simple? Since $x \ge 0$ it's $\frac{\pi}{2} \int_0^2 (\sqrt{(x-1)^2})^2\, dx = \frac{\pi}{3}.$
  12. Buzz Bloom

    Questions on Mass/Volume x Density: Exploring Nonlinear Effects in GR

    In another thread https://www.physicsforums.com/threads/why-does-the-universe-have-no-net-charge.843865/page-2#post-5326092 I was amazed to learn the following from a post by bcrowell: "You can't get the total mass of the system by adding up the masses of all its parts." The reason for this is...
  13. G

    MHB Volume of a solid of revolution

    Consider the solid in three dimensions that is formed when the graph of a function $f(x)$, with $f(x) \ge  0$ for all $x \in [a, b]$, is revolved around the $x$-axis on the segment $x \in [a, b]$. Derive the following formula for the volume $V$ of this solid: $V = \pi\int_a^b f^2(x)dx$. Use...
  14. S

    How to Calculate change in volume produced by a piston?

    Homework Statement Hello all! I need a little bit of help with my physics homework! It asks me to calculate the change in volume produced by the piston, and it gives me the equation: ∆V= Ax∆dHomework EquationsThe Attempt at a Solution Now I know that A= 1x 10-^16m^2 And ∆d=1700nm My only...
  15. Priyadarshini

    Chemistry Solving Al + BaNO3 Reactions for N2 Volume

    Homework Statement The reaction between aluminium powder and anhydrous barium nitrate is used as the propellant in some fireworks. The metal oxides and nitrogen are the only products. Which volume of nitrogen, measured under room conditions, is produced when 0.783 g of anhydrous barium nitrate...
  16. M

    Volume Charge Density of Proton Beam

    Homework Statement 1.0 mA proton beam accelerated through potential difference of 1 keV. Determine the volume charge density of the beam after acceleration assuming uniform current distribution within diameter of 5mm, with zero current outside of this. Particle starting from rest. Final answer...
  17. terryds

    Negative volume using washer method

    Homework Statement What is the volume of a solid formed by the area trapped between y= -x^2 and y= -2x rotated 360° around x-axis? Homework Equations V = ∫A(x)dx The Attempt at a Solution y=y -x^2 = -2x x^2 -2x = 0 x(x-2) = 0 This means that the two functions cross at x = 0 and x = 2 From x...
  18. S

    Volume in n Dimensions: Understanding the Meaning of n=0

    Hello, Surfing across the internet, I learned that the volume of a sphere in n dimensions can be expressed by V(n) = (Π^(n/2)) / Γ((n/2)+1), where n is the number of dimensions we are considering But if we consider n=0, then we get 1. So, how do we interpret this? I mean what does volume in zero...
  19. quaticle

    Finding volume of a submerged object

    Homework Statement There is a block of wood floating on the surface of a body of water, with a ball attached to the bottom of the block by a string. I am asked to find the volume of the ball given the tension in the string. We also know the volume of the wood block from an earlier problem if...
  20. 5P@N

    What % volume of this floating object is submerged?

    Homework Statement An object of 985 kg/cm^3 density is placed in water, which has a density of 1000 kg/m^3. What percentage of the object will be floating above the water? Homework EquationsThe Attempt at a Solution 985/1000 = .985, or 98.5%. 100 - 98.5 = 1.5%. Therefore: 1.5% of this object...
  21. K

    Did the Pressure of a Monatomic Gas Change During Isentropic Heating?

    Homework Statement A sample containing 3.65 mol of a monatomic ideal gas is heated from 289K to 458K, and the entropy remains constant. If the initial volume of the sample was 0.0980m^2, by what factor did the pressure increase or decrease during this process? Homework EquationsThe Attempt at...
  22. Hijaz Aslam

    Molar Specific Heat (gas) at varying pressure and volume?

    I've read in my texts that the there are two kinds of Molar Specific Heat Capacities for gases: 1. Molar Specific Heat Capacity at constant Volume ----- ##C_v## 2. Molar Specific Heat Capacity at constant Pressure ---- ##C_p## And in case of Constant temperature there is no point in...
  23. K

    What is the change in entropy ΔS of the gas?

    Homework Statement Two moles of an ideal gas undergo a reversible isothermal expansion from 3.37×10−2m3 to 4.29×10−2m3 at a temperature of 29.6 ∘C. What is the change in entropy ΔS of the gas? Homework Equations pV=nRT The Attempt at a Solution W=∫V2V1pdV, I don't know how to use this...
  24. T

    Volume of an octagonal dome by using calculus

    On this picture we see a octagonal dome. I am trying to calculate the volume of this object by integral calculus but I can't find a way. How would you calculate this? https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202015-12-17%20kl.%2002.14.48.png I am majoring in math-econ but...
  25. J

    Volume of a cylinder and radius

    Ok so i know the equation for the volume of a cylinder and the equation for calculating the radius. But when calculating the radius does the volume need to be converted into cubic inches or can it stay as imperial fluid ounces. Thanks
  26. U

    What is the maximum amount of light that can fit inside a mirrored sphere?

    I just joined to ask this question... I was imagining a sphere made of mirrors inside it and shooting a light to the inside (assuming none of it would escape back through the same hole). Once light enters, it will forever bounce inside the sphere. So my question is, will there be a limit of...
  27. F

    What happens to entropy when doubling the volume?

    Homework Statement A container of volume 2V is divided into two compartments of equal volume by an impenetrable wall. One of the compartments is filled with an ideal gas with N particles. The gas is in equilibrium and has a temperature T. How does the total energy, the entropy, the temperature...
  28. Stephanus

    Volume of 1 Mole O2 & CO2 at Same Temperature & Pressure

    Dear PF Forum, A: 1 mole O2 is roughly 32 grams? 1 mole ozone is roughly 48 grams? 1 mole CO2 is roughly 46 grams? Considering there are isotopes --------------------------------------------------------------- B: Do, at the same temperature and pressure, 1 mole CO2 and 1 mole O2 have the...
  29. 5P@N

    How do you calculate a human volume from a given weight?

    Homework Statement Hypothesize a human with a weight of: 61.14 kg (not an abnormal specimen in this wise). density of a human body: 985 kg/m^3Homework Equations What is the volume in cubic meters of such a hypothetical person? The Attempt at a Solution The mass of this person is determined...
  30. P

    Thermodynamics Control Volume evaluation 2 inlet

    Figure P6.95 provides steady-state test data for a control volume in which two entering streams of air mix to form a single exiting stream. Stray heat transfer and kinetic and potential energy effects are negligible. A hard-to-read photocopy of the data sheet indicates that the pressure of the...
  31. Eclair_de_XII

    How to find the volume of a square with function-based side?

    Homework Statement "[Find the volume of a] solid whose base is the region bounded by the curves (y = x2) and y = 2 - x2 and whose cross sections through the solid perpendicular to the x-axis are squares." Homework Equations A(f(x)) = f(x)2 V = ∫(A(f(x))dx Image of problem...
  32. CMATT

    Solving for volume and density

    <post removed from here by mentor, but copied to a later post>
  33. C

    What is the volume of the wood immersed in the water-filled flask?

    A piece of wood with a density of 730 kg/m3 is tied with a string to the bottom of a water-filled flask. The wood is completely immersed, and the tension in the string is 1.09 N. Find the volume of the wood.
  34. M

    Sphere Volume to Surface Area, Why not for Cone?

    Homework Statement Wikipedia tells me that I can obtain the surface area of a sphere by realizing that the volume of a sphere is equivalent to the infinite sum of the surface areas of hollow, nested spheres, sort of like little Russian dolls. That makes sense, and then differentiating both...
  35. B

    Constant Volume Calorimetry - Why does (ΔnRT)=(Δn)RT

    When doing constant volume calorimetry, the enthalpy can be calculated as follows: ΔH = ΔU + Δ(PV) ΔH = w + q + Δ(PV) ΔH = PΔV + q + Δ(PV) and at constant volume: ΔH = q + VΔP which I've then see people rewrite using the ideal gas law as follows: ΔH = q + (Δn)RT where Δn is the change in...
  36. D

    How is the inverse of a volume integral denoted?

    In 1-D the inverse of ∫ dx is dy/dx so how is the inverse of the volume integral ∫ d3x = ∫ dxdydz denoted ? Thanks
  37. P

    Finding volume of a modified sphere using integrals

    Hello guys, new member here. I've got a calculus project due Tuesday that I could use some help on. I won't bore you with the all details of the project, but first let's imagine an olive in the shape of a perfect sphere (with a radius always bigger than 6mm) that goes through a set of blades...
  38. M

    Ice-Cream Cone problem - Volume in Spherical Coord

    Homework Statement S is the sphere of equation x2 + y2 + z2 = 10z and C the cone of equation z= sqrt(3*( x2 + y2)) . The axes are measured centimeters. R of sphere = 5 D = 10 Total height is 10 cm Illustrate the solid E bounded by the C cone and the sphere S and calculate its volume using the...
  39. A

    Calculate variation in volume using given Poisson's ratio

    Homework Statement : A cylinder is elongated by 2% of its original length. If Poisson's ratio of its material is 0.3. Calculate the percentage variation in volume.[/B]Homework Equations V = πr^2l η= 0.3= Δl/l/Δr/r[/B]The Attempt at a Solution I tried using calculus to differentiate V to find...
  40. C

    Calculating Volume of Arbitrary Cross-Section Triangles

    Hello, I'm in the process of creating a calculation spreadsheet with Excel. In this spreadsheet I need to calculate a volume of an object that has an arbitrary cross-section but from the side of the cross-section, it is always triangle. At the moment the spreadsheet calculates the volume of a...
  41. K

    Ideal gas volume work expression (adiabatic)

    Homework Statement I have the following task: Homework EquationsThe Attempt at a Solution But I don't understand how to solve it. Can somebody help me?[/B]
  42. M

    I Question about converting propane vapor volume to liquid gallons

    Hello, I work for a propane company where i unload propane rail cars. I am trying to figure out how much propane is being shipped back to the refinery after I off load the rail car. We are experiencing a large amount of shink and the accountant want to get to the bottom of it. With propane...
  43. M

    Related Rates: Volume of Cone

    Homework Statement A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep. Homework EquationsThe Attempt at a Solution...
  44. M

    Related Rates of Volume Change for Expanding Cube Edges

    Homework Statement All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters? Homework EquationsThe Attempt at a Solution I used the equation for the volume of a cube: V = s3 but I'm not...
  45. A

    Calculating Volume Flow Rate of Olive Oil at a Processing Plant

    Homework Statement At a processing plant, olive oil of density 875 kg/m3 flows in a horizontal section of hose that constricts from a diameter of 2.92 cm to a diameter of 1.20 cm. Assume steady, ideal flow. What is the volume flow rate if the change in pressure between the two sections of hose...
  46. S

    Calculate volume from first order transfer function

    Homework Statement In a gas pipeline a mixture of hydrogen in air is transported. Usually, the hydrogen content of the gas is 1 volume %. It is well below the lower explosive limit as a hydrogen / air mixture is 4 % by volume hydrogen. gas pipeline is positioned a detector that activates an...
  47. J

    Finding volume ratio between 2 vessels using ideal gases

    Homework Statement I have been tasked with designing a feasible experiment to determine the ration between 2 vessels. I think i have a way that works on paper. Homework Equations pV = nRT and the conservation of mass. The Attempt at a Solution 1.Start with 2 vessels of unknown volume x and y...
  48. qq545282501

    Volume inside a hemisphere and a cylinder

    Homework Statement use spherical coordinates to find the volume of the solid inside the hemisphere z= √(25-x^2-y^2) and bounded laterally by the cylinder x^2+y^2=4 Homework Equations x=rcosθ =ρsinφcosθ , y=rsinθ =ρsinφsinθ z=ρcosφ r= ρsinφ The Attempt at a Solution I divided the solid into 2...
  49. Invutil

    Black Holes: Volume & Forces Explored

    Do black hole singularities have 0 volume? What forces are keeping the particles from being in the same place?
  50. M

    Calc II - Disk vs Shell method different volumes

    So I'm getting ready for an exam on tuesday, and I'm using each method for volumes of revolutions for every problem but I'm not getting the same answers. So, let's use this as an example: y = 5x; the shaded region is from [1,2] Using the disk method (about the x-axis) I find: R(x) = 5x; r(x)...
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