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. T

    How can you maximize the volume of a box made from a rectangular sheet?

    Suppose someone gives you a rectangular sheet of length a and width b (so b ≤ a). You make a topless box by cutting out a square with length x out of each corner and folding up the sides. How should you cut the sheet so as to maximize the volume of your box?
  2. AdityaDev

    Calculating volume occupied by 1 equivalent of gas

    Its given in my textbook that equivalence of gas = (volume of gas) / (Volume of 1 eq. gas at NTP) How do i calculate volume occupied by 1 equivalent of gas? Example: What would be the volume occupied by 1 eq. of CO2 (Its given 11.2l in my book) Is there a formula for this?
  3. Jess155

    Calculating Volume Charge Density: Simplified Earth Model

    Homework Statement In a particular region of the Earth's atmosphere, the electric field above the Earth's surface has been measured to be 148 N/C downward at an altitude of 260 m and 165 N/C downward at an altitude of 410 m. Calculate the volume charge density of the atmosphere, assuming it to...
  4. Togomori

    Discrete class D amplifier outputs low volume

    Hello everyone, I have a discrete class D project that I need help with but since I'm not an analog guy I got a problem. So far I got everything set up and the discrete class D works perfectly. But only at maximum of 6 volts otherwise the Transistor M3 gets really hot and burned, how do I solve...
  5. D

    Specific heat capacity and changing volume

    Hello everyone, I just need some help understanding some thermodynamics. So I have 0.25 kg of helium which is compressed from an initial state in a polytropic process with n = 1.3. So its given the change in volume and the initial pressure. I need to find the change in internal energy. I am...
  6. Dethrone

    MHB Calculating Volume of a Hollow Sphere w/ Differentials

    This is probably an elementary question, but I stumbled upon it while thinking about total differentials. One of their many applications is calculating the error in a volume, for example, given uncertainties in its dimensions. I'm not in the mood to tackle a 3D problem, so let's revert to a 2D...
  7. B

    Formula to find volume of a rectangular solid

    Hi all, I need help finding a formula to find the volume of a solid that has a fixed width and depth but 4 different heights. To see what I mean, please take a look at a sketch I drew below. Thanks.
  8. Mingsliced

    Thermodynamics: Gas pressure and change in volume

    Homework Statement Just want to check that I've used the correct method for this thermodynamics question I've been set. Any clarification would be greatly appreciated. So I have 0.5kg of gas held in a rigid container of volume 0.25m^3 at a temperature of 20°C. 20kJ of heat energy is required...
  9. Z

    How can I solve two related non-linear equations using EDP finite volume method?

    Hi evryone, the system i am working on, is composed of two relatied non-linear equations that I discretised using finite volume method with fully implicit scheme, the difficulty is that the two unknonw appear in both equations and i don't know how to solve it. Any propositions? Thank you.
  10. R

    Water hydro statics and presure

    hey guys. bellow is an upside down cone frustum with the dimensions given bellow where the green R is the radius of the larger circle and the blue r is the radius of the smaller circle and the red h to be the height of the object knowing that volume of this shape is calculated as...
  11. AdityaDev

    Finding the Centre of Mass of a Solid Cone

    Homework Statement Find the centre of mass of solid cone. Homework Equations $$y_{cm}=\frac{1}{M}\int_0^Hydm$$ The Attempt at a Solution First I took thin disks. I got the answer when I assumed its thickness to be dy but then dysecθ would be more accurate if half angle of cone is θ since...
  12. B

    Divergence and Volume Integrals

    Homework Statement (3 i) Using \nabla . \mathbf{F} = \frac{\partial \mathbf{F_{\rho}}}{\partial \rho} + \frac{\mathbf{F_{\rho}}}{\rho} + \frac{1}{\rho} \frac{\partial \mathbf{F_{\phi}}}{\partial \phi} + \frac{\partial \mathbf{F_{z}}}{\partial z} calculate the divergence of the vector field...
  13. A

    Problem with finding volume using shells

    A solid is generated by revolving the region bounded by y=(1/2)x2 and y= 2 about the y-axis. So what I did is set up my integral V= 2pi (integral sign from 0 to 2) x(1/2)(x)2 dx When I solve i get 4pi, and then I have to find the radius of the hole which has one fourth of the volume. So the...
  14. Matternot

    Volume of a quarter cylinder between 2 planes

    Homework Statement Find the volume V of the region that lies inside the quarter cylinder 0 ≤ r ≤ 1, 0 ≤ θ ⇐ 1/2 π and between the planes x+y+z=4 and z=0, where (r, θ, z) are cylindrical polar coordinates. Homework Equations integral dV = integral r drdθdz The Attempt at a Solution I...
  15. LiHJ

    What is the percentage of water in a sphere given its volume and radius?

    Homework Statement Dear Mentors and PF helpers, I can do part (a) and (b) but don't really know how to do (c) and (d). Can somebody teach me how to go about solving it. Homework Equations Volume of cone: $$\frac{1}{3}πr^2h$$ Volume of cylinder: $$πr^2h$$ Volume of sphere...
  16. pie

    Could a thing contain more than its volume can w/o overflowing?

    Hey, I am new in this forum and looking forward to learn physics which I hate :) My first question concerns possibility, in theory. Say, there is a water tank. Let us say that we made a nanometer hole. The tank is full. Then, we put even more water through that hole. But the hole has something...
  17. S

    Find the surface of the generated volume

    Homework Statement The region bounded by the graphs of the curves y=4√x , y=0, x=5, x=12 is revolved about the x-axis. a) Find the religion lateral surface area of the generated volume. b) Find the total surface area of the generated volume, including both ends. Homework EquationsThe Attempt at...
  18. V

    Finding two possibilities of depth when volume is given?

    Homework Statement Problem - If the volume of the carry-on luggage is 2000 cubic inches, determine two possibilities for its depth. Where necessary, round to the nearest tenth of an inch Homework Equations V(x) = x(x+10)(30-2x) Volume function is given above The Attempt at a Solution I...
  19. P

    Volume generated by 2sin(x/2) & sin(x) about y=2 with shells

    Homework Statement Find the volume generated by x=0 x=π y=2sin(x/2) y=sin(x) using the shell method. Homework Equations x=2arcsin(y/2) x=arcsin(y) The Attempt at a Solution So to setup my integral I got the following values, Outer Radius=(2-arcsin(y)) Inner Radius=(2-2arcsin(y/2)) I'm...
  20. N

    Gas laws: calculating total volume

    Homework Statement 5 lbs of nitroglycerin detonates. What is the total volume of the gases produced? Homework Equations PV=nRT The Attempt at a Solution I used the balance chemical equation of the decomposition of nitroglycerin and used stoichiometry to find the number of moles of nitrogen...
  21. A

    Looking for calculus program to find volume

    I need a website or a program that I can write an equation then find volume in Disk or washer or cylindrical and draw it ? it should be Free
  22. A

    Calculation of the force exerted by a liquid

    Question: The height of the given vessel is h,and the width of the given vessel is b (as given in the diagram). The density of the liquid is r.Find the force exerted by the liquid on the slant wall. Relevant formulae : P = F/A F = Vdg An attempt at the solution...
  23. Misha Kuznetsov

    Does ionization affect the volume of hydrogen gas at STP?

    At STP, a mole of hydrogen gas would take up 22.4 L. If that hydrogen gas was ionized and stripped of 100% of the electrons, would the remaining protons have a larger volume than 22.4 L? I'm not sure because on one hand, taking away the electrons leaves each proton with much less volume, but on...
  24. streeters

    Height of a spherical cap with known volume?

    Homework Statement What equation is needed to calculate the height of a spherical cap with a fixed volume and radius? Homework Equations V=πh/6 (3a^2 + h^2) Where V = volume, h is cap height, a is cap radius The Attempt at a Solution I have tried to separate the h out and got as far as...
  25. M

    MHB Volume of Liquid: Proving $\overrightarrow{v}\cdot\overrightarrow{n}$

    Hey! :o A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$. Let $\overrightarrow{n}$ an unit vector perpendicular to the plane. Show that $\overrightarrow{v} \cdot \overrightarrow{n}$ is the volume of the liquid that passes through the unit surface of...
  26. U

    Proper distance, Area and Volume given a Metric

    Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
  27. Pull and Twist

    MHB Solve Volume of Circle Rotated Around X-axis: Group Take Home

    I am working on a take home with a group of students and we all got different answers for this problem... can anyone look over it and tell me where I am going wrong?? Let me know if any of my work is confusing...
  28. kq6up

    Enthelpy for Constant Pressure and Constant Volume Systems?

    I understand why entropy (a state function) is very useful in chemistry. Since most chemistry systems are at constant pressure, ##\Delta H=Q_v## from the derivation below: ##H=U+PV## ##dH=dU+dPV+PdV## ##dU=\delta Q-\delta W## ## dU=\delta Q-PdV## ## dH=\delta Q+dPV## ##dPV=0## ## dH=\delta...
  29. G

    Effects of Intermolecular Forces?

    For a van der Waals gas, there are London dispersion forces causing the gas particles to be attracted to each other while the law of excluded volume provides repulsive forces. How do each of these characteristics effect the pressure, volume, and temperature of the gas? I assumed that for the...
  30. Pull and Twist

    MHB Need Help: Volume of Ellipse Rotated at X=-2 Using the Shell Method

    The question is... So of course I rearrange my ellipse formula to get y=\pm\sqrt{1-\frac{{x}^{2}}{4}} Then I calculate my radius as x-(-2)=r \implies x+2=r I know the formula for finding volume with the Shell Method is V=\int_{a}^{b}2\pi x\cdot f(x) \,dx To make it easier I decide to just...
  31. Drakkith

    Finding the Median Volume of a Sample in Chemistry Lab

    Homework Statement A question in my chemistry 151 lab book: A student measures the volume of water displaced by the same metal sample as 5.4 mL, 5.8 mL, and 5.5 mL for his three determinations. Determine the median volume. Homework Equations Unknown or N/A The Attempt at a Solution How do...
  32. M

    Tank Volume: Find Volume at 3,000 psi, 80 F

    Homework Statement A tank for scuba diving is designed to contain 50 standard cubic feet of air when filled to a pressure of 3,000 pounds per square inch (gage) at an ambient temperature of 80 F. Calculate the interior volume of the tank. A standard cubic foot occupies one cubic foot at T=15...
  33. A

    Gas Volume Calc: Find Out Rate of LPG Cylinder Reduction

    Hi can anyone please suggest me how to find out the rate at which gas in LPG cylinder reduce.Initially before use gas cylinder will be full as the consumer start using the gas it starts reducing how do i calculate how much is left out and the rate at which it is getting reduced ,And also if a...
  34. M

    Where Can I Find Resources to Help Me Solve Double Integral Problems?

    Homework Statement Find the volume of the solid bounded by the cylinder y^2+z^2=9 and the planes x=0, y=0, z=0, and 2x+y=2. Homework Equations None. The Attempt at a Solution This is double integral problem. I know how to find the double integral but I don't know how to set it up. How do I...
  35. T

    Volume of Rotation: Find Area Rotating Around Line

    Homework Statement Find volume when the area rotates around a line. Homework Equations x=y^2 x=1-y^2 about x = 3 The Attempt at a Solution Could someone please check my formula for Area? I have pi[ (3-(y^2))^2 - (2-(y^2))^2 ] which gives me the integral pi[5y-(2/3)y^3]. I think...
  36. T

    Volume Rot. Region: Find Vol. Bounded by Curve, Y=x^2 & x=y^2, Rotated about Y=1

    Homework Statement How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified lines? I keep getting a negative number and I"m becoming so frustrated! 11pi/30 is the answer in the textbook. Thank you so very much in advance...
  37. A

    Can a singularity expand in volume?

    Can a singularity expand in volume?
  38. samsara15

    Imaginary Volumes: Exploring Complex Numbers & Algebras

    Imaginary numbers enable one to envision a lot of ideas. But what kind of numbers/algebras would enable us to work with imaginary volumes? Volumes, by definition, always seem to be positive, since any cubes are. What kind of numbers would give/allow a more complex picture?
  39. Drakkith

    Which Object is the Most Dense Based on Given Measurements?

    Homework Statement Of the objects below, which is the most dense? A. an object with a volume of 13 dm3 and a mass of 1.29 × 103 g B. an object with a volume of 0.00212 m3 and a mass of 4.22 × 104 mg C. an object with a volume of 3.91 × 10-24 nm3 and a mass of 7.93 × 10-1 ng D. an object with...
  40. Christina H

    Calculations for cooling a volume of air

    Hallo, Working on building a cloud chamber based on thermoelectric coolers rather than dry ice or other "consumable" cooling methods, and having some difficulty working out some of the numbers. Say I have a volume of air at room temperature, for arguments sake 1600cm³, and I want to cool this...
  41. diegzumillo

    Entropy as log of omega (phase space volume)

    Homework Statement I've seen this problem appear in more than one textbook almost without any changes. It goes like this: Assume the entropy ##S## depends on the volume ##\bar{\Omega}## inside the energy shell: ##S(\bar{\Omega})=f(\bar{\Omega})##. Show that from the additivity of ##S## and the...
  42. C

    Pressure change when volume changes for incompressible fluid

    Homework Statement Hi guys, My first post here, I hope someone can help me out with a quick fluid mechanics problem. I'm looking to calculate the pressure increase inside a closed tube full of water when the tube is crushed and therefore its volume reduced. The tube is filled completely...
  43. H

    Thermal energy intensity per unit volume

    I am trying to figure out what the total thermal energy emitted by completely transparent matter. It seems Planck's law can't do this because the thickness or geometry of the mass is not given, only it's surface area. The spectral radiance of a black body is given by Planck's law...
  44. M

    What is the cause of an increase in volume (audio)

    This is just a question I posed to myself while learning about waves, I was wondering how the volume of an audible noise is increased or decreased, is it related to the compression of the wave and therefore the force generating it? If somebody could explain this in mathematical terms or provide...
  45. P

    Elastic balloon of volume V in vacuum

    Homework Statement Take an elastic balloon of volume V in vacuum. The surface of the balloon has tension T. Find the pressure inside the balloon in terms of V and T, then combine this to the ideal gas law to find an expression for V. Homework Equations See below The Attempt at a Solution I...
  46. M

    Volume flow rate and mass flow rate

    Hello there, so I have answered my question from fluid systems but the answer doesn't look right ;/ and I have no idea what have I done wrong. Can you please check and correct where it's wrong ? ;) here is the question: A pipe contains a gradually tapering section in which its diameter decreases...
  47. acdurbin953

    Relating Volume Charge Density and Linear Charge Density

    Homework Statement An infinite line of charge with linear density λ1 = 6.7 μC/m is positioned along the axis of a thick insulating shell of inner radius a = 2.4 cm and outer radius b = 4.7 cm. The insulating shell is uniformly charged with a volume density of ρ = -722 μC/m3. What is λ2, the...
  48. M

    Momentum/Impulse- Calculating Average Force of Raindrops

    Homework Statement The problem says: "A record rainstorm produced 304.8 mm (approximately 1 ft) of rain in 42 min. Estimate the average force that the rain exerted on the roof of a house that measures 10 m * 16 m. Indicate any assumptions you made. (Note: density of water is 1000 kg/m3"...
  49. L

    Molar heat capacity at constant volume

    Hi everyone, If you know the temperature rise of 2 moles of an ideal gas when a known amount of energy is transferred to it as heat, (hence are able to calculate cv by dU/dT); is the molar heat capacity simply half this value as it is half the number of moles?
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