What is Volumes: Definition and 192 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. Z

    Calculating Volume of Rotational Solids Using Integration

    Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y=x^2 y=2x about the x-axis. So i used intergral 0 to 2 (2x-x^2)^2 dx this is wat i got... pi*(((2^5)/5)-(4(2^4)/4)+(4(2^3)/3)) its wrong so wat did i do wrong?:confused:
  2. X

    Volumes of Revolution Word Problem

    Homework Statement Assume that the Earth is a sphere with circumference of 24,900 miles. a. Find the volume of the Earth north of latitude 45 degrees. (hint: integrate with respect to y) b. Find the volume of the Earth between the equator and latitude 45Homework Equations circle: x^2 + y^2...
  3. A

    Volume of Region A in First Quadrant Rotated Around y = -2

    [SOLVED] Volumes of Revolution Homework Statement find the volume of region A in the first quadrant that is inclosed in the parabola 2x(2-x) and the x-axis, of which is rotated around the axis y = -2. Homework Equations y=(4+2x^2) y= -2 piS(from 0 to 2) (R^2 - r^2) The Attempt at...
  4. Q

    Volume Calculation Using Cylindrical Shells

    Consider the given curves to do the following. x = 3 + (y-2)**2, x = 4 Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. V = ?? **************** I set up the problem like this... V...
  5. A

    Interested in finding volumes with multivariables to understand the background

    I can't find the volume of solid sqrt(x) + sqrt(y) + sqrt(z) = 1. It's a graph but I wish I had a graphing calculator to see it. It's bounded by x=0, y=0, z=0. I'm teaching myself this stuff and think integration using a change of variables by making x=u^2? This would be a transformation of T...
  6. O

    Area and Volumes of Solid of Revolution

    1. find the area common to r=1+cos@ and r=3^(1/2) sin@ 2. find the volume generated by rotating the region bounded by (x-1)^2 + (y-2)^2 = 4 around a. x axis b. y -axis c. x = 3 d. y = 4
  7. M

    Proving Areas, Surfaces, and Volumes Using Integral Methods

    hi how can the following be proved using integral methods: a) prove surface area of sphere, radius a, is 4 \pi a^2 b) prove area of a disk, radius a, is \pi a^2 c) prove volume of ball, radius a, is \frac{4}{3} \pi a^3 d) prove volume of axisymmetric cone of height h and base with radius...
  8. F

    Learning to solve for volumes? help appreciated.

    Hello Everyone, I am a bit new here, and I hope this will be the correct area to ask for help on this. (I've been away from math for a good number of years). I am trying to figure out how to find the volume of an area removed from a cylinder. Say I have a cylinder with a radius of 6...
  9. S

    Volumes of revolution not around the axis

    Homework Statement Find the volume of y = 2x^2 y = 0, x = 2 when it is revolved around the line y = 8.Homework Equations Integral formulas for volumes by discs, washers and cylinders.The Attempt at a Solution Translate the curve so that axis of revolution is along the X axis. Is this the...
  10. A

    Volumes, applications of integrations

    find the volume of the solid by rotating the region bounded by the given curves about the specified line, y=x^4, y=1; about y=2 how do i set up the problem so i can figure out the area, i don't need the answer, and i already graphed it, and i already rotated the graph,
  11. M

    Comparing Volumes of Gases in 2 Cylinder Containers

    I want to ask if I have 2 cylinder container 1) 50 m3 2) 60 m3 If I inject the same no. of mole of gases into these containers respectively when I apply PV=nRT Is the volume in calculating the gases in these containers are the same??
  12. D

    Determining best method for volumes?

    Homework Statement No specific question but what is the best way to determine which method is the best for solving for volumes (using integration) of shapes as they revolve around axis/lines. disk method? washer method? shell method? other? what exactly do you look for that may hint...
  13. C

    Calculus NoviceEvaluating Washer/Shell Volumes - Calculus Novice

    Hi there, I'm new here and I'm really glad that I found a community to discuss calculus even though its not really my favorite course (although I enjoy some of its topics). My first question here is about evaluating volumes of shapes that have no solid interiors. I know how to use the...
  14. C

    Calculating Volume of Revolution: Bounded Figure Rotated about y = -1

    What is the volume of the figure bounded by y = 2x - x^2, y = 2x, x = 2, and rotated about the line y = -1. Is this the correct integral? \[ V = \pi \int_0^2 {((2x + 1)^2 - (2x - x^2 + 1)^2 } )dx \] Thank you for your time.
  15. T

    Need help with volumes of spheres with holes.

    Homework Statement if you guys have trouble reading that, it says, Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure...
  16. U

    Calculating Volume of Solid Revolution around x = 1

    Find the volume of the solid generated by revolving the region bounded by y = 2x - x^2 and y = x about the line x = 1 Ok, so I can solve this if it were revolved around the y-axis, it would be: \pi \int_0^1 (x)^2 - (2x - x^2)^2 dy If I have to revolve it around x = 1, what do I need to change...
  17. U

    Calculating Volume of Revolutions in a Glass Chamber with Water Using Equations

    There is a glass chamber that is to be filled with water. The chamber is divided into two parts, the outer being filled with water and the inner being empty. The chamber is such that a person can stand inside without getting wet. It is modeled by the equations f(x) = \frac{243}{1820x^2} -...
  18. K

    Help! Solving for Volumes of HNO_2 and NaNO_2

    Wow, I am having a brain fart :(. I can't seem to figure this question out. Here's some information that I've figured out: pH = pKa + log (\frac{[NO_2^-]}{[HNO_2]}) 3.00 = 3.40 + log (\frac{[NO_2^-]}{[HNO_2]}) \frac{[NO_2^-]}{[HNO_2]} = .40 Also, pH = -log[H^+] [H^+] = .0010 M...
  19. Hootenanny

    Calculating the Volume of Revolution for e^x + 1 on x-axis

    If the finite region bounded by the curve y = \text{e}^{x} +1 , the y-axis and the line x = \ln2 is rotatated around the x-axis by 360^{\circ} show that the volume of the solid formed is: \frac{\pi}{2} (7 + \ln4 ) I did the intergral and got: V = \pi \left[ (\text{e}^{4} + 2\text{e}^{2}...
  20. T

    Calculating Cork Volumes for Amusement Park Water Slide

    An amusement park is planning to build a water slide, and the park owners would like to use giant corks with density 0.28 g/cm^3 as "floats" for young children. What volume of cork is needed to keep 20% (by volume) of a 40 kg child above the water at the base of the slide (assuming the density...
  21. Q_Goest

    Tools for Consciousness, Control Volumes

    This thread contains section 2 and part of section 1 of a 5 section paper I'm writing regarding consciousness. Section 2 is intended to create a more general theory around something used in engineering called "control volumes". I've also included part of section 1, the abstract, to provide a...
  22. N

    Volume & Area: Can a Man Lift 10x His Weight?

    People, suppose a man can lift a weight equal to his own weight 2 mtrs. off the ground with very little effort. If his dimensions (i.e. LENGTH, BREDTH, HEIGHT) are increased 10 times, keeping his average density constant, will it be easier for him to lift his new weight 2 mtrs. off the ground...
  23. S

    Quantum Volumes and Observables: Exploring Macroscopic States in Quantum Physics

    Marcus has Highlighted an interesting paper:http://www.arxiv.org/abs/gr-qc/0509049 But I think it should be here in the Quantum Physics thread, and it definately should have some discussion? Quote here from the paper: The first one is what observables (besides the volume) should be...
  24. T

    What is the ratio V2/V1 of the volumes?

    I need help with this question: Sphere 1 has surface area A1 and volume V1, and Sphere 2 has surface area A2 and volume V2. If the radius of Sphere 2 is six times the radius of Sphere 1, what is the ration A2/A1 of the areas? Part two: What is the ratio V2/V1 of the volumes? :confused:
  25. bayan

    How to Use Integration to Find the Volume of a Rotated Function in Mathematica

    hi felles. I am trying to find what is the volume of the y=\frac{a}{x^2}+b is when it is rotated in y-axis. The values of a is 1 and b is -1. max hight is 3 and min is 0. I was trying to integrade and ended up with V=\frac{-Pi}{y^2+2Y+1} where y is 3. Is this right? I did a U...
  26. V

    Calculating the Intersection Volume of Two Intersecting Cylinders

    the question i have in my assignment is this : "Find the volume common to two circular cylinders each with radius r if the axes of the cylinders intersect at right angles" now, I'm not even sure what they're asking, let alone where to begin. not looking for answers, just a nudge in the...
  27. Q_Goest

    Terminology for Control Volumes, Elements & Nodes

    I'm looking for some terminology. Numerical methods of analysis seem to use different terminology for the same things, such as an "element" or "finite element" used for stress analysis, versus a "control volume" in thermodynamics. I've also seen both terms used in CFD. Is there a proper term...
  28. K

    Washer method and disc method for finding volumes of graphs

    Hi, I was wondering how do you know which method to use when let's say, they give you two equations and say to find the volume of the solid rotated on a certain axis. Is there a certain rule of thumb to follow (like in stock market - buy low sell high?)? I am really confused. Thanks in advance.
  29. A

    Volumes in the 4th spatial dimension

    How would you calculate the volume of a 4-dimensional object? Like a hypercube, hypersphere, etc...
  30. H

    Software for Calculating Volumes of Revolution on Linux, Mac, Windows

    I need software that can calculate volumes of revolutions, the only requirement is that it runs on any of the following: * Linux (i386) * Mac OS X * Windows XP (i386) Like calculate the volume when tan x is rotated around the y-axis with upper boundaries 0.2 and lower boundraries 0.1.
  31. B

    How Is Volume Calculated Using Cylindrical Shells?

    Hi, I was hoping someone could check my work on a few problems and get me started on a few others. It involves definite integration, so I'm going to use (a,b)S as an integration symbol and P for pi. These are the ones I need checked: 1. Use cylindrical shells to find the volume of the...
  32. M

    Solving Pressure Shooter Kink: Find Volume & Air Needed

    I've made this pressure shooter for school, but I can't quite figure out this kink. If the cylinder has a volume of 4,994.57 cubic centimeters, and I plan on filling it with 100 psi of air (I want to maximize the pressure) how much air am I going to need? I could just flow it in, but I need to...
  33. G

    Finding volumes of solids involving exponentials

    I'm attempting to find the volume of the solid obtained by rotating the region under the curve: e^{-x^2} Bounded by y = 0, x = 0, and x = 1. I've done quite a few of these problems before, however, none of them have involved an exponential. If I recall correctly e^{x^2} cannot be...
  34. L

    Calculate the volumes of air and hydrogen gas

    If anyone knows how to do this problem, could you please give me some hints?? I don’t need the answer just some help as to where to start... Metallic molybdenum can be produced from the mineral molybdenite, MoS2. The mineral is first oxidized in air to molybdenum trioxide and sulfur...
  35. K

    Directionality in Stokes Theorem for Volumes

    I'm not sure if this post should go here or in the Calc setion, but I figure more knowledgeable people browse this form. This question is relating to 'directionality' of doing closed loop integrals. If you have some 2D wire structure, let's image it looks like a square wave, or a square well...
  36. G

    Archived thread Volumes of Regular Icosahedron and Regular Tetrahedron

    Archived thread "Volumes of Regular Icosahedron and Regular Tetrahedron" Hi, The above-referenced thread is at this url address: https://www.physicsforums.com/archive/t-3876 I have something to add, having worked with this structure, in a bit of a different way, however, than is spoken...
  37. O

    Finding Volume by Revolving Curves: Troubleshooting the Washer Method

    I'm having trouble coming up with the correct solution to a number of these questions.. This is one I've been stumped on.. Find the Volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis. y = sqrt(25-x^2), y = 3 Since I believe it...
  38. J

    How to Calculate the Rate of Change of Surface Area for an Expanding Sphere?

    How do you get the volume of an expanding object? For example: A sphere's is increasing at 12ft/sec, the volume is 36, how fast is the surface area of a sphere V=4*pi*r^3/3 S=4*pi*r^2 dV/dt= 12 So i took the derivative of the volume function dV/dt=4*pi*r^2*dr/dt 12=4*pi*r^2*dr/dt...
  39. G

    Volumes Generated by Revolving the Area Bounded by x=y^2 and x=4

    Can someone please help me w/ these problems below: Find the volume generated by revolving the area abounded by x=y^2 and x=4 about a)the line y=2 b)the line x= -1 ***I tried to write out the integral not sure if it's correct: a) V=pi* int. of (sqrt(x)^2-(2-sqrt(x))^2) dx **integral form...
  40. G

    Calculus Help: Find Volumes & Solve Problems

    Calculus help please! Can someone please help me to figure out how to do these problems below?...Or at least get me started in the right direction. Thanks a lot for ur help!I would really appreciate ur help! 1) Find the volume of the solid of revolution generated when the area bounded by...
  41. S

    Chemo Reacting Volumes Of Gasses q's

    Hi guys i need a bit of help, with a couple of questions, if possible can someone show me how to work these with the answer, thanks a lot, any help appreciated! 1.A mixture of 2 litres of methane (Ch,4) and 4 litres of oxygen was ignited causing a combustion. Calulate the composition and...
  42. J

    Volumes of Regular Icosahedron and Regular Tetrahedron

    Please teach me. Is one Regular Icosahedron equal to twenty Regular Tetrahedrons ? If the edgelength of both Regular Polyhedras is 1, What would be their volumes ? Can we prove (or disprove) the equation below ? volume of Regular Icosahedron = 20 * volume of Regular Tetrahedron...
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