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. A

    MHB Find the volume of the solid of revolution, or state that it does not exist. #2

    I'm having some trouble with this problem: Find the volume of the solid of revolution, or state that it does not exist. The region bounded by f(x)= 6(4-x)^(-1/3) and the x-axis on the interval [0,4) is revolved avout the y-axis. How would I be able to tell whether to use the shell, disk, or...
  2. A

    MHB Find the volume of the solid of revolution, or state that it does not exist.

    Find the volume of the solid of revolution, or state that it does not exist. The region bounded by f(x)= the square root of ((x+3)/(x^3)) and the x-axis on the interval [1,infinity) is revolved around the x-axis. I tried using the disk method: pi* (sqrt(((x+3)/(x^3)))^2 Then I think I have to...
  3. T

    Volume of a solid using disks/washers

    Homework Statement Find the volume of the solid generated by rotating the region enclosed by y=\frac{1}{1+x^2} , x=-1,x=1 and y=0 about the line y=2 Homework Equations pi(outer radius)^2-pi(inner radius)^2 The Attempt at a Solution Since i am rotating around a horizontal line i figured...
  4. E

    Calculate the Volume of a Lemonsqueezer

    Homework Statement f(x)=\frac{1}{81}*x^4-\frac{5}{9}*x^2+4 The tangent in Point P(6|0) when rotated around the y-Axis gives the Shape of the Squeezer. The bottom is at y=-5, the top at y=0 The Attempt at a Solution First I calculated the tangent and got t: y=4x-24 Then I converted that to...
  5. J

    Area and volume calculation (no integration))

    I can compute the area of the rectangle formed by Δx and Δy simply by product ΔxΔy. Now, how can I to compute the area in gray given Δr and Δθ? Also, I can to compute the volume of a parallelepiped formed by Δx, Δy and Δz, simply multiplicand ΔxΔyΔz. But, how can I compute the volume...
  6. reenmachine

    Random question about cones and cylinders volume

    A cone's volume with height ##x## and radius ##y## is ##1/3## of the volume of a cylinder with height ##x## and radius ##y##.I was trying to visualize it in my head and struggled a bit.Take a rectangle triangle with height ##x## and the other side of length ##y## which isn't the hypothenuse ...
  7. A

    How Does Reorienting a Cylinder Affect the Juice Level?

    A cylindrical container of height 1 m and diameter 0.5 m is partially filled with apple juice. When the container is lying on its side, the juice level at the deepest point is 37.5 cm (three eighths of a meter from the bottom of the cylinder is full). What is the liquid level after the container...
  8. R

    Calculating Volume of a Double-Lobed Cam Using Polar Coordinates

    Homework Statement The surface of a double lobed cam are modeled by the inequalities: \frac{1}{4}\leqr\leq\frac{1}{2}(1+cos2θ) and -9/(4(x2+y2+9)) ≤ z ≤ 9/(4(x2+y2+9)) Find the volume of the steel in the cam. Homework Equations The Attempt at a Solution I know I...
  9. R

    Calculating Volume of Tetrahedron Using Triple Integral: Step by Step Guide

    Homework Statement Set up an integral to find the volume of the tetrahedron with vertices (0,0,0), (2,1,0), (0,2,0), (0,0,3).Homework Equations The Attempt at a Solution My method of solving this involves using a triple integral. The first step is deciding on the bounds of the triple integral...
  10. V

    Volume charge density w/o surface charge density

    Im confused by a concept i have run across in Griffiths electrodynamics. E_{out} - E_{in} = \frac{\sigma_{free}}{\epsilon_0} However, in the case of a uniform, circular charge density, \vec{E_{in}} = \frac{\rho r}{3\epsilon_0}\hat{r} \vec{E_{out}} = \frac{\rho R^3}{3\epsilon_0...
  11. D

    Last edited by a moderator: May 6, 2017

    Hello Homework Statement Show that for an ideal gas: n(E)dE=2πn/(kπT)3/2 *E1/2 exp(-E/kT) dE where n(E) is the number of particles for each element of volume whose energy is between E and E+dE Homework Equations The Attempt at a Solution Really don't know where to start...
  12. B

    Le chatelier, pressure and volume

    This question has been bugging me and the more I think about it the more confused I get. N2O4 ⇔2NO2 Question: the reaction will shift to the right with all of the following changes except A. Addition of N2O4 B. an increase in volume at constant pressure C. A decrease in pressure at...
  13. J

    Calculating Volume Using the Disk Method for Revolving Regions

    Homework Statement Find the volume of the solid generated by revolving the region bounded by the parabola y=x^2 and the line y=1 about the line y=1 Homework Equations V= integral of pi*r^2 from a to b with respect to variable "x" The Attempt at a Solution pi(integral of 1-(x^2-1)^2...
  14. P

    Finding the volume of air in a box when it's lowered into water?

    Homework Statement A box that is open at the bottom is lowered into the sea (density like water). The outer volume of the box and the air inside it is V_{out}=3 m^3. The moment the box touches the sea surface the air inside it gets trapped and has a volume at V_0=2.5 m^3 and a pressure at...
  15. J

    Volume of solid rotated around y=1

    Homework Statement Find the volume of the solid formed by revolving the region bounded by f(x) = 2-x^2 and g(x) = 1 about the line y = 1. Homework Equations V = ∏∫(1-f(x))^2dx - ∏∫(1-g(x))^2dx The Attempt at a Solution I keep ending up with ∏∫(1-(2-x^2))^2dx - ∏∫(1-1)^2dx, on...
  16. V

    Volume Integral Orthogonal Polynomials

    Hello. Homework Statement Basically I want to evaluate the integral as shown in this document: Homework Equations The Attempt at a Solution The integral with the complex exponentials yields a Kronecker Delta. My question is whether this Delta can be taken inside the integral...
  17. Saitama

    MHB How to Find the Volume of a Tetrahedron?

    Problem: Suppose in a tetrahedron ABCD, AB=1; CD=$\sqrt{3}$; the distance and the angle between the skew lines AB and CD are 2 and $\pi/3$ respectively. Find the volume of tetrahedron. Attempt: Let the points A,B,C and D be represented by the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$...
  18. T

    Solving for the Volume of a Solid Using Double Integrals

    Homework Statement Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8. Homework Equations V = ∫∫ Height Base The Attempt at a Solution I first found...
  19. J

    Area element, volume element and matrix

    I found this matrix in the wiki: https://fr.wikipedia.org/wiki/Vitesse_ar%C3%A9olaire#.C3.89valuation_en_coordonn.C3.A9es_cart.C3.A9siennes I think that it is very interesting because it express d²A not trivially as dxdy. So, I'd like of know if exist a matrix formulation for volume...
  20. L

    MHB Maximizing the volume of a beam cut from a cylindrical trunk

    what are the dimensions of rectangular beam of volume maximum that can be cut from a trunk in diameter "D" and length "L", assuming that the trunk has the shaped of a straight circular cylinder shape? Answer Width =lenght
  21. L

    MHB Maximizing the volume of a cylindrical postal package

    The sum of the length and the perimeter of base of a postal package to is 60 cm. find the maximum volume: when the package is cylindrical. The answer is 2547 cm3 V cilinder = pir2h and the sum L + L+H = 60 2L + H = 60 solving for H and putting it into the volume i don't get the answer Yeah...
  22. D

    MHB Maximizing Volume of a 5-Sided Box w/ Cutout Corners

    Consider a sheet of length L and width W. Each corner is cut out (x by x corners removed). Detemine the value of x so when the corners are removed and flaps folded up, the five sided box formed will have maximum volume. SA \(= 1LW + 2 LH + 2WH\) and V \(= LWH\). I am not sure how to do this...
  23. L

    MHB Thank you for your understanding.

    A tree trunk is shaped like a truncated cone it has 2 m of length and diameters of their bases are 10 cm and 20 cm. Cut a square straight section so that the axis of the beam coincides with the axis of the truncated cone. find the beam volume maximum that can be drawn from this form. answer...
  24. D

    MHB Volume of a Pyramid: Find A(z) to Calculate V

    I am trying to find the volume of a pyramid where the base has length \(L\) and width \(W\), and the pyramid has height \(h\). Let \(L\) be on the x-axis and \(W\) be on the y axis. In the x-z plane, we have the line \(z = -\frac{h}{L/2}x + h\), and in the y-z plane, we have the line \(z =...
  25. Y

    Volume of a solid with 3 boundary conditions

    Homework Statement Find the volume of an object bounded by x2 + y2 ≤ 1, x2 + z2 ≤ 1 and y2 + z2 ≤ 1. Homework Equations The Attempt at a Solution This stuff is very new to me (multiple integrals to find volume) so I am not entirely familiar with it. My first thought was to put...
  26. R

    Cylindrical shells to find volume of a torus

    Homework Statement Use cylindrical shells to find the volume of a torus with radii r and R. Homework Equations V= ∫[a,b] 2πxf(x)dx y= sqrt(r2 - (x-R)2) The Attempt at a Solution V= ∫ [R, R+r] 2πx sqrt(r2 - x2 - 2xR + R2) dx I feel like this isn't going in the right direction...
  27. L

    Energy needed to push a volume of water

    Hello! I just found this website and it looks amazing! I'm not a scientist or anything, but I love it (should've studied physics but oh well), so I think it will be fun and useful for me to join this forum. I am trying to solve a situation, where I'd like to know how much energy would be...
  28. H

    Using Gauss's Law to find E for an infinite volume charge density

    My E&M professor brought up this problem of considering a uniform charge density, rho, that is infinite in volume and then using Gauss's Law to find the electric field at a point. It's resulted in a lot of head scratching and I'd appreciate some help/discussion to guide me towards a resolution...
  29. C

    Density, volume and buoyancy

    1. At Lagoon, there is a large granite rock ball that is supported by water pressure, so people can spin the rock. The diameter of the rock is 1.3m. Granite has a density of 2691kg/m^3. Let’s assume a water pressure if 50 lbs/in^2. Calculate the area of the ball that must be in the water...
  30. rsyed5

    MHB Max Volume: Finding Constraints, Dimensions

    So, I have this question, but I have no idea what constraint is and how to find a constraint for the length, height and width... and if i say the square wastage is x, then the width is 80-x but I don't know what the length would be with respect to x... , and how do we determine the dimensions..?
  31. M

    Show that the Change in Volume is Independent of the Path

    Homework Statement Homework Equations The Attempt at a Solution I understand what the question is asking. Both ways I should get the same answer. I'm having trouble figuring out the mathematics behind this question.
  32. S

    Help with Volume of Revolution/Trig Substitution Problem

    Homework Statement The problem is attached in this post. Homework Equations The problem is attached in this post. The Attempt at a Solution Disk method with the radius equal to x/((x^2+3)^5/4) For Trig Substitution √(x^2+a^2) -> x=atanθ a=√3 -> a^2=3 x=√(3)tanθ -> dx=√(3)sec^2(θ)...
  33. MarkFL

    MHB Nick's question at Yahoo Answers regarding a volume by slicing

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  34. C

    Volume of Solid of Revolution for y=x^2-2, y=0 about y=-1

    Homework Statement Find the volume of the solid of revolution obtained by rotating the area bounded by the curves about the line indicated. y=x2-2, y=0 about y=-1. Need only consider part above y=-1 Homework Equations V=∏a∫b[f(x)]2dx The Attempt at a Solution I'm mainly unsure of...
  35. pellman

    The invariant momentum-space volume element?

    When we way that \frac{d^3p}{p_0}=\frac{d^3p}{\sqrt{m^2+\vec{p}^2}} is the invariant volume element, is that with respect to all Lorentz transformations or just proper orthochronous Lorentz transformations?
  36. MarkFL

    MHB Nick's question at Yahoo Answers regarding a volume by slicing

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  37. S

    Help with volume of solid of revolution/integration by parts question

    Homework Statement The problem is attached in this post. Homework Equations The problem is attached in this post. The Attempt at a Solution I've set up the integral via disk method: π∫(e^√x)^2 dx from 0 to 1 I've done integration by parts by don't know how to integrate the...
  38. A

    What volume does the v in pv denotes?

    what volume does the "v" in pv denotes? say that in a system where pressure is constant Mg reacts with O2.when dealing with above reaction thermodynamically, HI=UI+PVI where H is the initial enthalpy of the system UI is the...
  39. C

    MHB Rotation around a curve. Find the Volume.

    I am thinking about how to find the volume rotate around its function.Let f be a function of x in the interval [a,b] . The function could be any curve. And the curve is rotation around itself. Would there exist a volume of the curve? And how to find the volumeThank you CBARKER1
  40. M

    Finite Volume Method & Evaluating Integral at Borders for Two-Phase Flow

    Hi! I am trying to make a one-dimentional simulator for two-phase flow. I am going to use the finite volume method, because it is conservative and thus it's easier to keep track of the oil/water ratio in the area. Say you have a conservation equation on the form \nabla \cdot (k(x) \nabla P(x))...
  41. O

    Can the volume of primitive unit cell and unit cell be different?

    Hi all, I read The unit cell is the smallest structure that repeats itself by translation through the crystal. Some says premitive unit cells contains atoms only at the corners while a unit cell may contain extra atoms in between(like bcc or fcc). At one place I found this: For each...
  42. cjweber

    Qualifying Propylene Glycol amounts for specific volume

    Hello all, I am a newbie to this site and have found some interesting discussions herein, so I thought it worthwhile asking the collective wisdom of this group about Hygroscopic liquid calculations that I am struggling to correlate. It has been several years now since I worked as an electronics...
  43. S

    MHB Finding the volume of regions rotated about the x - axis

    Need someone to verify that my work is correct please. Consider the region bounded by $y = sin(x)$ and the x - axis from $ x = 0$ to $x = \pi$ a) Find the volume if the region is rotated about the x - axis. V = \int \pi (sin(x))^2 \, dx \pi \int^{\pi}_0 sin^2x \, dx \pi \int^{\pi}_0...
  44. N

    Determining volume of fluid dispersed from a broken pipe

    I have the pipe size, flow rate, and a duration. How can I figure out the amount of fluid that was released from the pipe?
  45. L

    MHB Maximizing the volume of a cone formed by revolving a right triangle

    A triangle hypotenuse given rectangle is rotated around one of their legs to generate a right circular cone? find the cone of greater volume. resp V= (2Sqrt(3)pi L^3)/27 It says hypotenuse given but it has no value According to the answer you can name it L
  46. L

    MHB Maximizing the volume of a cone formed by cutting a sector from a circle

    From A circular sheet of RADIUS "R" a sector tie is cuts so that the coil Gets a funnel. Calculate the angle of the circular sector to cut back so of funnel has the maximum capacity. Answer tha angle is 2sqrt(6)pi/3
  47. L

    MHB Maximizing the volume of a buoy

    From two places equal of radio circular R It wants to build a buoy consists of two equal bass common cones Determine the radius of Ia base when the volume of the buoy is maximum. r=(sqrt of 6) R/3
  48. MarkFL

    MHB OliverG's Calculus Q: Estimating Pool Volume w/ Trap Rule

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  49. A

    MHB What are the methods for finding the volume of a solid of revolution?

    i have no clue as to how to proceed for the following three problems: #1 find the volume of the resulting solid by any method x^2 + (y - 1)^2 = 1, about the y -axis #2 use the cylindrical method to obtain the volume of a sphere of radius r #3 and a right circular cone of radius r and height h...
  50. S

    Help with solid of revolution volume question

    Homework Statement The problem is attached in this post. Homework Equations The problem is attached in this post. The Attempt at a Solution I used washer method and set my outer radius as 2+2+√(x-1) and my inner radius as 2. I set my upper limit as 5 and my lower limit as 2...
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