Volumes Definition and 192 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. Feodalherren

    Why Does Turpentine Not Overflow When Heated in an Aluminum Cylinder?

    Homework Statement A hollow aluminum cylinder 2cm deep has an internal capacity of 2L at 20C. It is completely filled with turpentine and then slowly warmed to 80C. a) How much turpentine overflows? b)If the cylinder is then cooled back to 20 C how far below the cylinder's rim does the...
  2. M

    Volumes of Solids of Revolution

    Question I'm really having issues grasping the Volumes of Solids of revolution. I could use some help solving this question, it isn't very hard. 1. Let R be the region bounded by y = x2 and y = x + 2. Find: a) the area of R b) the volume of the solid if R is rotated about the...
  3. S

    MHB Volumes by Cylindrical Shells (Calculus II)

    Quick question, may seem rather dumb - but I just want to make sure of something.. Question: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2 so when I am setting up my integral am I correct in saying TOP - BOTTOM i.e. --> \int^1_0 (2\pi...
  4. M

    About Compressive Forces on Control Volumes

    Why is it that forces (for changes in pressure across a fluid) on the control volume are generally considered compressive? Even in the derivation of the Navier-Stokes equation, it is assumed that forces from fluid pressure will be compressive? Why?
  5. MarkFL

    MHB Jon feafe's questions at Yahoo Answers regarding volumes by slicing

    Here are the questions: I have posted a link there to this thread so the OP can view my work. edit: This question has since been deleted at Yahoo! Answers.
  6. MarkFL

    MHB Airshow4444's question at Yahoo Answers regarding volumes by slicing

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  7. MarkFL

    MHB On the volumes of pyramids

    This topic is for questions and comments pertaining to this topic: http://mathhelpboards.com/math-notes-49/volumes-pyramids-6131.html
  8. L

    Percetnage by volume of propanol from original volumes

    if i had 500ml of water and added 250ml of propanol, how would I go about working out its percentage by volume? thanks
  9. R

    Mass and energy analysis of control Volumes (thermodynamics)

    Homework Statement 1) A cylinder with an initial volume of 1.5 m3 initially contains water 1 MPa and 200 ° C (condition 1). Container is cooled at constant temperature until the volume is 42% of the initial volume (state 2). The constant temperature process is followed by a constant volume...
  10. E

    Volumes of Solids with Known Cross Section Project

    I'm trying to get started on this project but am totally confused about how to find the volume of the solid. All the information I was given was the following: y= √x boundaries: 0,9 cross sections: isosceles right triangle how the hell do I get started?!
  11. G

    Volumes with triple integrals, aka I suck at geometry

    Homework Statement Calculate the volume of the body that is bounded by the planes: x+y-z = 0 y-z = 0 y+z = 0 x+y+z = 2 Homework Equations The Attempt at a Solution I made a variable substitution u = y+z v = y-z w = x which gave me the new boundaries u+w = 2...
  12. J

    Statics, centroids of lines, areas and volumes

    Hi to everybody. I´m reading a book about statics and I cannot understand this chapter. I have been calculating moments of forces in hundreds of problems, when I found a force acting on a body I needed to fix a coordinate system, then calculating the moment arms of that force around a point...
  13. H

    Find Volume of Solid with Trapezoid Cross Sections

    A solid has as its base the region bounded by the curves y=-2x^2=2 and y=-x^2 +1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis is a trapezoid with lower base in the xy-plane upper base equal to 1/2 the length of the lower base, and height equal to 2...
  14. Petrus

    MHB Calc Volumes of Rotation Bodies | x-axis & 7x-x^2

    Calculate the volumes of the rotation bodies which arises when the area D in the xy-plane bounded by x-axis and curve 7x-x^2may rotate around x- respective y-axes. I will calculate V_x and V_y I start to get crit point x_1=0 and x_2=7 rotate on y-axe: 2\pi\int_a^bf(x)dx so we get...
  15. E

    Finding Volume Using Integration: Revolving Functions Around a Line

    Homework Statement Find the volume of the solid formed by revolving the functions: y=x2 and y=1 about the line y=6 Homework Equations ∏∫(ro2-ri2)dx The Attempt at a Solution I found the outer radius to be (1-x2)-6 and my inner radius to be -6. Also, the limits of integration were...
  16. D

    Gaseous Chemical Reactions and Volumes

    (Hi everyone! Apologising for the trivial(and likely boring) question in advance. Sadly, it has me boggled for some reason). Homework Statement In a certain temperature and under a pressure, 500 mL of H2, and 100 mL of O2 are poured into a container. A Chemical reaction occurs as follows...
  17. D

    Calculating Volume of Overlapping Regions using Integration

    Homework Statement THe region bounded by y = -x + 3 and y = x^2 - 3x the region revolve about a, x-axis, and b, y=axisHomework Equations V = π∫r^2 dxThe Attempt at a Solution I have no clue to solve it since the volume overlap. I try to ignore the overlapped region but didn't get the right...
  18. L

    Finding volumes via double integrals

    Homework Statement Find the volume which lies below the plane z = 2x + 3y and whose base in the x - y plane is bounded by the x- and y-axes and the line x + y = 1. Homework Equations I = \int\int_{R} f(x, y) dydx = \int^{b}_{a}\int^{y=y_{2}(x)}_{y=y_{1}(x)} f(x, y) dydx The...
  19. S

    Boyles Law to Calculate Lung Volumes

    Hi All, This is a bit long, so only read on if you have too much time on your hands... Clinically its important to know how large someones lungs are. Asking them to breathe out as much as they can, will not tell us the lung volume, as there is always a residual volume of gas left in the...
  20. H

    Production of large volumes of duterated hydrocarbons?

    Hi everyone, I'm a nuclear engineering student so my chemistry knowledge is limited to a couple of courses in first year so please be patient with me. I was wondering how difficult or expensive would it be to produce hydrocarbons with deuterium instead of hydrogen on a large scale? (say...
  21. I

    What is the volume of these round pipes?

    Hi How much volume would be in the following round pipes 1. 1000mm x 50mm 2. 1000mm x 76mm 3. 1000mm x 101mm 1 liquid has a S.G = 1 1 liquid has a S.G = 1.30 Any help appreciated IJC7
  22. T

    Volumes by Slicing and Rotation about an Axis.

    Homework Statement Find the volume of the solid generated by revolving the region between the parabola x = y^2 + 1 and the line x = 3 about the line x = 3. Homework Equations The answer is found by integrating with respect to y with disk method, but I don't understand why my answer is...
  23. ShayanJ

    Gauss's law and volumes with zero «net» charge

    The integral form of gauss's law is used to determine the electric field of charge distributions which possesses a certain amount of symmetry. Now imagine using it in situations where the gaussian surface includes equal amounts of positive and negative charge. For example,imagine a point...
  24. H

    Scaling Polynomial Functions for Water Bottle Design

    Homework Statement For an assignment, I'm required to design three water bottle using 3 different polynomial functions. I've used a linear, quadratic and cubic. The first bottle needs to be 600ml, the second 300ml, and the third 1L. In order to 'create' the bottles, we are to calculate the...
  25. J

    Volumes of Solids of Revolutions Help

    Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y=x, y=0, x=2, x=4; about x=1 The Attempt at a Solution one of the radii is x=y but I am...
  26. A

    MHB Finding volumes by multiple integrals

    How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me. Find the volumes in R3. 1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1. 2. Find the volume W that is bounded by the cylindrical...
  27. C

    Exterior algebra and n-dimensional volumes

    Hello, In R^3, the surface of the parallelogram determined by two vectors u and v is given by the norm of the cross product of u and v. For my research, I have to know if this can be generalized in the following manner: Let e_1,..,e_n be the canonical basis of R^n, and Ext_k be the exterior...
  28. M

    Calculating Volumes: Proving Cylindrical Shell Disks

    I am trying to find proofs on the internet of how cylindrical shells disks work but I can't find any. Are there any proofs out there?
  29. T

    Simple calculus volumes integration

    Homework Statement Find the volume of this equation, revolved around x axis Homework Equations y=x^2 y^2=x The Attempt at a Solution 1) (pi)(r^2) 2) r = x^2 3) (pi)((x^2)^2) 4) (pi)(x^4) now to integrate 5) (pi)(1/5(x^5)) since x = 1, and 1^5=1, 1/5=1/5 pi/5? there...
  30. Hunus

    Question on Landau and Lifshitz volumes

    What are the mathematical prerequisites of these books? In particular, what are the mathematical prerequisites of volume I?
  31. C

    Thermodynamics: two pistons; different pressures, volumes, and temperatures

    Homework Statement Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed as shown in the figure. One vessel of volume V1 = 15.2 L, contains oxygen at a temperature of T1 = 280 K and a pressure of P1 = 1.77 atm. The other vessel of volume V2 =...
  32. 2

    Help extending volumes of revolution to fourth dimension

    I am currently learning about volumes of revolution in calulus, and have looked ahead to surfaces of revolution as well. I want to try and extend this concept to revolving 3d functions over the x-axis into the fourth dimension. I found this thread...
  33. E

    Accurately dispense liquid volumes

    I am considering the economic viability of a project that must accurately (within a tolerance of, say, 0.5ml) dispense volumes of water from a tank at a constant volumetric flow rate of around 50ml/s. Within the tank, the water may be held at pressures between 1 and 6bar; and at temperatures...
  34. I

    Volume of cathedral dome (Using volumes of revolution, disk method)

    Homework Statement A cathedral dome is designed with three semi circular supports of radius r so that each horiontal cross section is a regular hexagon. Show that the volume of the dome is r^3 * sqrt(3) an accompanying figure - http://imgur.com/3fSqh Homework Equations...
  35. B

    Calculating Volumes to Prepare Standard Solutions with KSCN

    Calculate the volume of each reagent required to prepare 25.0 mL of each standard solution at the specified KSCN concentration. Fill in the rest of the table (attached). My complete guess is that the solution II needs to be incremented by 2.5 ml each time.
  36. M

    Volumes of Rotating Functions: Shell vs. Washer Method

    Is there a general rule when to use the shell or washer method when working with calculating volumes of functions rotating about an axis? For instance should I use the shell method when rotating about the y-axis and use the washer method when rotating about the x?
  37. J

    Volumes By Cylindrical Shells (2 questions)

    Homework Statement HEY EVERYONE! Sorry for putting up another quesiton, but I'm almost done my assignment for the week. I understanding volumes by cylindrical shells, but i get confused when rotating about a line that's not the x or y axis. I have uploaded the two questions. Homework...
  38. M

    Currently my class it calculating volumes of solids by rotating them

    Currently my class it calculating volumes of solids by rotating them about some axis, say for instance the function f(x) = x^2 bounded by s = { (x,y) | 0≤x≤1 , 0≤y≤1} and rotating it about the y - axis. I understand the general look of the graph on paper but I can't visualize the actual solid...
  39. D

    Double Integral volumes: Triangular base

    Evaluate ∫∫R 5x2 + 2y2 where R is triangle (1,1) (2,0) (2,2) I see the lines bounding the triangle are y=x y=2-x and x=2, and have tried many attempts at setting up the correct limits. Would it be correct to split this into 2 triangles, or are the limits y=x∫y=2-x for y and 2∫1...
  40. gmax137

    What happens to the units of L as n approaches infinity in hypercube volumes?

    I am reading Julian Havil’s book Nonplussed, and in one chapter he’s discussing hypercubes, he says that the volume of an n-dimensional cube of side length L is L^n; then he goes on to note that as n-> infinity, the volume goes to zero if L<1; volume goes to 1 if L=1, and volume goes to infinity...
  41. P

    Solid State Hydrogen storage volumes

    I am trying to figure out what the actual efficiency of solid state hydrogen storage is. So how many kg of hydrogen can you store in a Xcm3 of metal hybride solid state storage. I was trying to find a conversion and only found a metric that was 500NL in a specific device that had some...
  42. B

    Volumes question, volume of a torus

    Problem solved I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :( Homework Statement Let a and b be constants, with a > b > 0.A torus is formed by rotating the circle (x - a)^2 + y^2 = b^2 about the y-axis. The cross-section at y =...
  43. I

    Volumes with Cylindrical Shell Method

    Homework Statement Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y= 4x2, y=-2x+6 Homework Equationsy= 4x2, y=-2x+6 These 2 equations meet at x= -3/2 and x=1 integral from a to b of...
  44. J

    Calculating volumes by shell and slicing

    Homework Statement The area under the graph of the function y = cos inverse x on the interval [0; 1] is rotated about the x-axis to form a solid of revolution. (a) Write down the volume V of the solid as a de nite integral with respect to x according to the disc/slicing method. Do NOT...
  45. E

    How can we use calculus to find volumes of revolution?

    As part of an assignment on Approximating Areas and Volume I am asked to derive the equation shown in the image attached. The question reads: "It can be shown that if y = f(x) is revolved around the x-axis to form a solid between x=a and x=b then the volume of the solid is give by the...
  46. J

    Volumes of solids of revolutions

    Homework Statement In the question your working with the region bounded by the two curves y=x^2 and y=x^3. In the first part of the question you had to revolve the region around the x-axis and find the area which I managed to do by subtracting two areas from each other. The second part of...
  47. K

    Volume Slicing: Identifying Annulous Solids of Revolution

    Most of the time I can visualise whether some solids of revolution are annulous or not but sometimes I just don't see it. Can anyone tell me if I am missing something? Is there any way of knowing if the cross section is annulous? Please help...
  48. R

    Potential Energy of a Liquid - Why is it Negative for Small Molar Volumes?

    I was just reading a book on low-temperature physics and stumbled upon a graph that shows the energy of liquid Helium-4 in relation to its molar volume. The graph includes both zero-point energy and the potential energy of the liquid, and the latter goes steeply (basically a vertical line) from...
  49. A

    Volume of Revolution: Cylinders vs Washers

    Homework Statement Find the volume of the solid bounded by the curves y = x^{1/3} and y = x when rotated around y=1. Homework Equations Volume with washers: V = \pi \int R(x)^{2}-r(x)^{2} dx where R(x) and r(x) are functions of x defining the inner and outer radii of the washers...
  50. K

    Thermodynamics, two equal volumes of gas at STP mix, entropy change?

    Homework Statement Calculate entropy change if two equal volumes of gas (1 cm3) at standard temperature and pressure are separated from each other by a partition which is removed to create twice the volume at standard temperature and pressure Homework Equations dS = k.N. dV/V + C/T dT...
Back
Top