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. J

    Washer vs cylindrical shell method for computing volumes

    Hello, Homework Statement My problem regards the disk|washer, and cylindrical shell methods for finding volumes in single variable calc. My problem is basically am I understanding these two methods and their relationships properly. Fundamentally, these methods are indentical, as we can...
  2. agnibho

    This is a problem on mensuration ( VOLUMES AND SURFACE AREAS)

    Homework Statement A right circular cone of diameter r cm and height 12 cm rests on the base of right circular cylinder of radius r cm. their bases are in the same plane and the cylinder is filled with water up to a height of 12 cm. If the cone is then removed, find the height to which water...
  3. M

    I keep getting negative volumes (volume of sin(x) around y=c, c is in [0,1])

    Homework Statement The arch y= sin(x), x is in [0, pi], is revolved around the line y=c, where c is a constant in [0, 1], to generate a solid... Anyway, then I have to represent the volume of the solid as a function of c and other stuff. Homework Equations The Attempt at a Solution...
  4. P

    Calculating Volume of Wedge Cut from Maple Tree | Integration Problem

    Hi, I am having a lot of trouble with this problem. Homework Statement A lumberjack is preparing to cut down a large maple tree. With a chain saw, he makes a horizontal cut exactly halfway through the trunk, and then makes a second cut at 45 degrees, meeting the first cut along the...
  5. D

    If we take bigger and bigger volumes

    (Holographic principle) Volume of a sphere is proportional to R^3 However, max amount of information inside is proportional to it's surface, to R^2 So information density is proportional to 1/R It means that if we take bigger volumes, the content inside appears to be correlated with the...
  6. A

    Single Var Calculus - Volumes of Revolution

    Single Var Calculus -- Volumes of Revolution consider the curves y = 6 x = 0 and y = x2+2 Revolve the bound area around the y-axis and find the volume of the product solid. Here's what I did. r = x = (y - 2)1/2 V = pi * INTEGRAL from y = 2 -> y = 6 of r2 dy = y - 2 dy = y2/2 -...
  7. J

    Finding Extreme Volumes of V(x): A Box Model

    Homework Statement The function V(x) = x(14-2x)(15-x) 0<x<7 models the volume of a box Find the extreme values of V and tell whether these represent the smallest and largest volumes or only the smallest or only the largest volumes Homework Equations The Attempt at a Solution...
  8. D

    Volumes of a Region bounded by Two Curves

    1. Let R be the region bounded by the curves y = x2 and y = x + 2. (a) Sketch the region R and label the points of intersection between the two curves. (b) Suppose we rotate R about the x-axis. Compute the volume of the resulting solid. (c) What is the volume of the solid obtained by rotating...
  9. B

    Energy Analysis of Control Volumes at Steady State

    Homework Statement Steam enters a well insulated nozzle at... Pressure1 =300lbf/in^2 Temp1 =600 degrees F Velocity1 =100 ft/s The steam exits the nozzle at... Pressure2 =40lbf/in^2 Temp2 =? Velocity2 =1800ft/s For steady-state operation, and neglecting potential energy effects...
  10. Shackleford

    Infinitesimal areas and volumes for common structres

    We pretty much do derivations maybe 80% of the time in my Intermediate Mechanics class. I'm having a bit of trouble seeing the various infinitesimal areas or volumes when incorporating that into an infinitesimal mass and density equation in our gravitational chapter we're in right now. Is there...
  11. romsofia

    Quantum Mechanics- Albert Messiah (Two volumes bound as one)

    Any reviews of this book? Any would be helpful!
  12. Z

    Standard errors in surface areas and volumes?

    I have to finish this one question that I have come across and I am having a bit of trouble figuring out where to go from what I havee done. The Q is: A copper cylinder is 5.82 +/- 0.06 cm long and has a radius of 2.53 +/- 0.04 cm. Using the appropraite formula, Question Details a) Find...
  13. S

    Finding volumes by rotating around an axis of revolution

    1. Homework Statement k so here is the equation i need help with that will find me the volume of a sphere 2*pi*y*sqrt(25-(y-1)^2) dy - 5*pi from 0 to 6 the 5 pi is the volume of a cylinder 2. Steps so first i subbed u=y-1 took the 2 pi out of the integral that got me 2 integrals...
  14. K

    Drag/friction on a cylinder rotating in water of different volumes

    This is not a homework question, but a tipsy bar question.The question being: There are two tanks of water, one large and one small, full of water of the same density. You are using engines to rotate one cylinder in each tank, the cylinders being of equal size. Will the cylinder in the larger...
  15. S

    Finding Volumes by using the Disc and Washer Method

    Theres a few key concepts about the disc and washer method that I can't quite grasp and I was hoping if I could get a bit of clarification. 1) How do you find your outer and inner radius? I can provide an example if needed. 2) If a problem has its function, for example f(x)= sec x...
  16. A

    Help with setting up Integrals for volumes

    Homework Statement Hi I am trying to find the volume of a shape.I don't need help to solve, but I would like a hand setting these integrals up. I've only recently started doing volumes, so bear with me. Using the curves: y=x x=2-y^2 y=0 Indicate the method used and set the...
  17. A

    Volumes of Cross Sections with Perpendicular Planes

    Hey guys I am new here i was wondering if anyone can help me understand this problem better Homework Statement The base of a certain solid is the circle x^2 + y^2 = 9. Cross sections of the solid with planes perpendicular to the y-axis are semicircles with their diameter in the base of the...
  18. F

    Volumes of Solid of Revolution

    I have a volume problem that I has been bothering me for a while now, as I have just not been able to figure it out. The question involves finding the volume of the solid generated by revolving the region bounded by y = x, y = 0, and y = 4 around the line x = 6. I tried doing \int ^{4}_{0}16...
  19. H

    Filling different shaped volumes

    Ok, I need some math info and I think this is the right section for my question, but I may be wrong. Is it physics? General math? I don't know. I'd like to know if you can determine the shape of a volume by the rate differences with which it fills up. Let's say you are filling a ball with...
  20. K

    Volumes of solids of revolution

    Homework Statement A circle with a radius of (a/2) is bored through the centre of a sphere of radius a. Find the volume of the remaining solid. Homework Equations The Attempt at a Solution I've been trying this for an hour now and I've been trying to find the remaining volume by...
  21. P

    Parametric Equations and integrals that represent volumes

    Homework Statement A surface S is formed by rotating a quarter ellipse C about the X-axis. Write an integral that represents the volume enclosed by S. the ellipse is represented by two points, (2,1) at which t= pi/2, and (4,0) at which t=0. Homework Equations Ellipse w/radii a,b, in...
  22. B

    Volumes of revolution

    R(x)=x^3 bounded by x=0, x=2 and y=1. a. revolved around x=2 b. revolved around x=10 my pathetic attempt: a. v=pi[(integral from 0 to 1)(2-y^1/3)^2]dy so =pi[4y-3y^(4/3)+(3/5)y^(5/3)]evaluated from 0 to 1 =(8/5)pi b. v=pi[(integral from 0 to 2)(10-x^3)^2]dy I must admit that I...
  23. P

    What is the Volume of the Solid Generated by Revolving a Triangular Region?

    Homework Statement Find the volume of the solid generated by revolving the triangular region bounded by the lines y= 2x, y= 0 and x= 1 about the line x= 1. Homework Equations V= \int A(x)dx = \int \pi[R(x)]^{2}dx The Attempt at a Solution I used the disk method, in which I found...
  24. S

    Calculating Volume of Revolved Solid Using Cylindrical Shells

    This shouldn't be a problem but I just can't this right! Homework Statement Find the volume of the body that is generated when y= 1/x(sqrt(4-x^2)) ,1 <= x <2, is rotated around the y-axisHomework Equations Using cylindrical shells : 2 pi ∫x (g(x)-f(x)) dx The Attempt at a Solution 2 pi ∫x...
  25. P

    Finding the Volume of a Wedge Using Calculus - Calculus Worksheet Solution

    A log having the shape of a right circular cylinder of radius a is lying on its side. A wedge is removed from the log by making a vertical cut and another cut at an angle of 45 degrees, both cuts intersecting at the center of the log. Find the volume of the wedge. And as soon as I finish...
  26. L

    How to Calculate Water Level Changes in a Filling Trough

    Homework Statement A trough in the shape of an isosceles trapezoid is 30 cm wide on the bottom, 80 cm on the top, 50 cm tall, and 10 cm long. It is being filled with water at 0.2 cm3/min. How fast is the water level rising when the water is 30 cm deep. The Attempt at a Solution Well I...
  27. P

    Integrate Volumes: Revolve Region Bounded by x=0, x=1, y=0, y=x^5

    Homework Statement Revolve the region bounded by x=0, x=1, y=0 and y=x^5 about the y-axis use shells to find the volume I know how to set up the integral I just don't know where I'm integrating from. Is it from 0 to 1?
  28. H

    Calculating O2 and Bi2S3 for Volume Law of Combining Volumes

    Homework Statement Consider the reaction 2Bi2S3 +9O2 ---------> 2Bi2O3 +6O2 The reaction is carried out and kept at 1.00 atm pressure and 174 C throughout. How many liters of O2 are needed to produce 3.87 L of SO2? How many grams of Bi2S3 are needed? The Attempt at a Solution my...
  29. T

    Volumes by Slicing and Rotation About an Axis

    Homework Statement I'm having issues with this section in calc. I'm not at all sure what I'm doing! Here is the problem I'm having trouble with: Directions: Find the volume of the solids. Problem: The base of the solid is the disk X^2 + Y^2 <= 1. The cross-sections by planes...
  30. C

    Volumes of Spheres around a Box

    Homework Statement Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L,W, and H. Homework Equations 4/3(pi)r The Attempt at a Solution I have gotten to...
  31. N

    Volumes of Revolution - Ellipsoid

    Homework Statement An ellipse is rotated around the y-axis, find the volume of this solid. Homework Equations x^2 / a^2 + y^2 / b^2 = 1 \pi\int_{a}^-a x^2 dy The Attempt at a Solution I'm having trouble solving this; I know that the upper and lower bounds of the curve occur on...
  32. B

    Surface areas and volumes

    can anyone give derivations for formulas for finding volume,curved surface areas,total surface areas..of a frustrum...based on similarities of triangles... And anyone belonging to gurudatta coaching centre please add me to skype adikap5735
  33. P

    General Equations for Certain Volumes of Revolution

    The volume of a cone = 1 - B H where B is the base of the cone and H is its height. 3 We can think about a cone as the line y = x rotated with respect to the y axis. The...
  34. B

    Triple Integrals (volumes)

    Homework Statement A solid is definited by the inequalities 0\leqx\leq1, 0\leqy\leq1, and 0\leqz\leqx2+y2. The temperature of the solid is given by the function T=25-3z. Find the average temperature of the solid. The Attempt at a Solution I solved the integral, however I could not...
  35. D

    Find Volume of Revolution by Integrating 1/sqroot(3x+2) around x=0 and x=2

    so i have y = 1/sqroot(3x+2) find volume when rotated around x, regions are x=2 and x=0 equation needed: V= integral Pi*y^2*dx so. i do intergral pi* (1/sqroot(3x+2))^2 * dx so i get pi integral 1/(3x+2) dx so how do i integrate 1/sqroot(3x+2) ? can someone take me...
  36. U

    Calculus project: volumes, rates, etc

    Homework Statement Houdini is in a giant flask and he stands on a block where his feet are shackled and you need to calculate some vital information to help him out. Cross section of flask = r(h)=10/(sqrt(h+1)) water is being pumped into the flask at 22pi Takes Houdini 10 mins to escape Ignore...
  37. O

    Titration of HBrO: Calculating pH of KOH Volumes

    Homework Statement Consider the titration of 20.0mL of a 0.100 M solution of HBrO, a weak acid (Ka=2.5x10-9) with 0.200 M KOH. Calculate the pH of the following volumes of KOH. a. 0.00mL b. 5.00mL c. 10.00mL d. 30.00mL Homework Equations pH=pKa +log [base]/[acid] The Attempt...
  38. S

    Motion of Volumes of Air in a Wave-Filled Tube

    Homework Statement Imagine a row of many small volumes of air (each consisting of many billions of molecules) along the entire length of a tube that is open at one end. Describe the motion of these small volumes of air when the standing wave, represented by Fig2 (sound wave in a closed tube)...
  39. T

    Finding volumes of revolution using centroids

    In my Cal II, we're discussing finding volumes of revolution using centroids, which we find using moments of x or y. Can someone explain to me what a moment is?
  40. T

    Calculating volumes by shell and disc method (Looking for Professional advice)

    Homework Statement The function is y = 2 - x. The region is bounded by x = 2 and x = 4. Calculate its volume by the shell method by rotating it by the x axis. The Attempt at a Solution This problem has been consuming my mind. I calculated it by the disc method and shell method but I...
  41. W

    How can I calculate the work for different volumes of cylinders underground?

    Homework Statement http://www.math.rutgers.edu/~greenfie/mill_courses/math152/pdfstuff/w2.pdf problem 2 Homework Equations Work is the integral of force.. The Attempt at a Solution Problem 2: Basically I know how to calculate the work for the cylinder. Since they have...
  42. E

    Finding Volume Limits and Integration for Paraboloid of Revolution

    Volumes: The Disk Method [Resolved] 1. If the area bounded by the parabola y = H - (H/R^{2})x^{2} and the x-axis is revolved about the y-axis, the resulting bullet-shaped solid is a segment of a paraboloid of revolition with height H and radius of base R. Show its volume is half the volume of...
  43. L

    Volumes of Revolution: Disk vs. Shell Method Explained

    I've encountered a weird problem in my text...somewhat by accident =P My text only covers volumes of revolution through the disk method, and one of the questions was: Find the volume of the solid obtained when the given region is rotated about the x-axis. c) Under y = 1/x from 1 to 4 Using...
  44. B

    Volumes of Revolution: Revolving about a vertical axis

    Homework Statement Find the volume of the solid obtained by rotating the region under the graph of y=1/x and y=1/x^2 about the vertical axis x=-1 Also given are points on the y-axis (0,2) and (0,5). I guessing these points are specific sections of the graphs where we find the volume...
  45. S

    Question on volumes of water, excess mass, potential energy and power

    Ive done this question but from the answers I am getting I can see that I am going wrong, but I don't know where. There are lots of different parts to this question, I will show them all, followed by my attempt at working them out. A Harbour has an average water covered area of approx...
  46. rocomath

    Check , Volumes for Calculus 1

    Check please, Volumes for Calculus 1 Been a while since I've done these type of problems, check please! http://img70.imageshack.us/img70/3558/volumes1iy0.jpg" http://img70.imageshack.us/img70/6768/volumes2aa9.jpg" 1. Find the volume of the solid generated by revolving the region about the...
  47. W

    Question about general theory of areas and volumes

    This is an elementary question: restricting ourselves to the euclidean plane, is there a strict definition of what kind of set of points constitutes a region with area? For example, does a set of points describing a circle adjoined with an isolated point outside the circle still constitutes a...
  48. rocomath

    Calculating Prism Length Using Integration: Strang Ch. 8, Example 5

    I have a simple question on the set-up of a triangular prism. Strang, Ch. 8: Applications of the Integral pdf page 4 (bottom) and correlates with example 5 on pdf page 5 http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/8.1-8.3.pdf How is he getting the length of 1-\frac x h
  49. P

    What's Next in Calculus: Solving Volumes with Integrals

    [SOLVED] volumes... last week of calc i was absent when we went over the volume section. y= 1/x, x= 1, x= 2, y= 0,; about the x-axis \int^{1}_{2} \pi \frac{1}{x} dx i don't know what's next. can anyone inform me, please
  50. N

    Finding volumes from infinitesimal displacements

    Homework Statement In spherical polar coordinates, the infinitesimal displacement ds is given by: ds^2 = dr^2 + r^2 d\theta ^2 + r^2 \sin \left( \theta \right)^2 d\phi ^2 Can I find the volume of a sphere using ds? The Attempt at a Solution I know the spherical volume-element is given...
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