What is Volume of solid: Definition and 77 Discussions

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.
Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid Theorem).
A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length w) around some axis (located r units away), so that a cylindrical volume of πr2w units is enclosed.

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  1. S

    Volume of solid having triangle base and semicircle slice

    First, I tried to find the equation of line passing through (2, 0) and (0, 3) and I got ##y=3-\frac{3}{2}x## Then I set up equation for the area of one slice, ##A(x)## $$A(x)=\frac{1}{2} \pi r^2$$ $$=\frac{1}{2} \pi \left( \frac{1}{2}y\right)^2$$ $$=\frac{1}{2} \pi...
  2. PeroK

    I Volume of Solid of Revolution (About the line y = x)

    I found this problem, which I thought was interesting and somewhat original: Calculate the volume of the solid of revolution of the area between the line ##y = x## and the parabola ##y = x^2## from ##x = 0## to ##x = 1## when rotated about the axis ##y = x##.
  3. brotherbobby

    Dropping anchor from the barge

    (I must confess before I can begin that I found this problem difficult to understand, for reasons I will make clear below. I know it appears simple.) Attempt : Let me begin by drawing the problem situation alongside, to the best I understand. We can see that the in both cases (i) or (ii), the...
  4. W

    Volume of solid region double integral

    I sketched this out. With the z=0 and y=0 boundaries, we are looking at ##z \geq 0## and ##y \geq 0## I believe ##0 \leq x \leq 5## because of the boundary of ##y=\sqrt{25-x^2}##. This is my region ##\int_0^5 \int_0^\sqrt{25-x^2} x \, dydx ## ## =\int_0^5 xy \vert_{0}^{\sqrt{25-x^2}} \, dx##...
  5. O

    Double integral, find volume of solid

    Homework Statement Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders: y = 1 − x2, y = x2 − 1 and the planes: x + y + z = 2 4x + 5y − z + 20 = 0 Homework Equations ∫∫f(x,y) dA The Attempt at a Solution So I solved for z in the plane...
  6. C

    Volume of solid by using projection to different planes

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  7. C

    MHB  Volume of solid in first octant

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  8. S

    Negative Volume of Revolution?

    There is a nice equation made by Nobuo Yamamoto which describes the curve of an egg and it is: (x^2 + y^2)^2 = ax^3 + (3/10)xy^2, where a is the length of the major axis of the egg. Solve this equation for y, we get: y=+/- sqrt((3/20)ax - x^2 + xsqrt((7/10)ax + (9/400)a^2)) When I rotate the...
  9. M

    Finding this volume without cylindrical shells

    This is not a homework problem this is just a problem I was thinking about whether or not it would be possible to solve without cylinderical shells The region bounded by y=1/(x^2) x=e x=e^3 and the x-axis Rotated around y axis.. I did cylindrical shells and got 4pi and then wondered if I...
  10. L

    Finding volume of a nose cone with a given r with integration

    I'm still confused on some of these volume problems, so please bear with me :) Homework Statement Find the volume of a reentry spacecraft nose cone that has a cross-section radius of (1/4)x2 taken x feet from the nose and perpendicular to the axis of sym. We are given that the length of...
  11. B

    Volume of solid x^2 + (y-1)^2 =1 about y-axis

    Homework Statement Hello, I am to find the volume of the solid given by x2 + (y-1)2=1 rotated about the y-axis. I may use either shells or cylindrical method. I attempted shell method, but am just learning this, still foggy and this is the one question that isn't coming out right. Homework...
  12. K

    MHB Volume of Solid of Revolution; Simpson's Rule

    1.find tthe volume solid generated by revolving the region bounded y=sqrt x and the ;lines y=1, x=4 about the line y=1 2. using simpson rule witj n=4 to aproximate int from 0 to 1 1 over 1-x power 2 dx
  13. veronica1999

    MHB Volume of Solid with Circular Base & Equilateral Triangles

    A solid has a circular base of radius 3. If every plane cross section perpendicular to the x-axis is an equilateral triangle, then it's volume is I keep on getting 18 root 3. But the answer is 36 root 3. Could I get some help? Thanks.
  14. 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...
  15. 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...
  16. 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...
  17. sheldonrocks97

    What is the Volume of the Solid Using Cylindrical Shells for y=-e^(-x^2)?

    Homework Statement Find the volume of the solid using cylindrical shells: y=e-x^2 y=0, x=0, x=1, about y-axis. Homework Equations How do I integrate xe^(-x^2)? The Attempt at a Solution 2∏x∫0 to 1 xe^(-x^2) dx 2∏*-(e^(-1))/2)
  18. C

    Integral Calc: Volume of Solid of Revolution

    Homework Statement Find the volume of the first quadrant region bounded by x=y-y3, x=1 and y=1 that is revolved about the y-axis. 2. The attempt at a solution v=∏ ∫ from 0 to 1 of (y-y^3)^2 dy and doing this, I got the answer to be 8∏/105. Did I set up that integral...
  19. M

    Volume of solid of revolution - y axis.

    Homework Statement Find the volume of the solid of revolution when we rotate the area limited by the x-axis and the function f(x) = 1 - cosx where x e [0, 2∏] once around the y-axis? The Attempt at a Solution In my notes I have the following equation: V = ∫ 2∏x f(x) dx If I put...
  20. P

    Calculating Volume of Solid Using Triple Integral

    1. Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder x^2 + z^2 = 4 and the planes y = -1 and y + z = 4 This looked like a cylindrical coordinate system to me, except for the fact that it is not cylindrical around the z-axis but the y-axis. I...
  21. L

    MHB Find volume of solid generated (Calc 2)

    [solved]Find volume of solid generated (Calc 2) Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y=e^x, and the line x = ln 2 about the line x= ln 2. So I tried graphing it to see visually, and the expression I got...
  22. I

    Using double integration in finding volume of solid bounded by curves?

    Homework Statement The question is "Use double integration to find the volume of the solid bounded by the cylinder x2+y2=9 and the planes z=1 and x+z=5" Homework Equations The Attempt at a Solution I tried to draw the curves and the solid that i formed is a cylinder with a...
  23. J

    Not understanding what the question is asking. Volume of solid

    Homework Statement Referring to the parallelepiped in Question 3 in which vertices E,F,G,H are respectively the diametrically opposite corners to A,B,C,D . Find the volume, in cubic units of the solid with seven plane faces BGD,BCD,GFD,BHG,GHEF,EFGC,BHEC Homework Equations The...
  24. P

    Volume of solid x=1+y^2 using shell method

    Homework Statement Volume of solid rotated about x-axis x=1+y^2, y=1, y=0, x=0 using shell method Homework Equations The Attempt at a Solution so i set up the integral with ∫2pi(y)(1+y^2)dy from 1 to 2 which is apparently wrong, but i don't know why.
  25. M

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    Homework Statement Consider the region bounded by the curves y= lnx and y=( x-3)^2 Find the volume of the solid obtained by rotating the region about the y-axis Homework Equations The Attempt at a Solution For this I solve for the x so i got x= e^y and x= (y)^(1/2) +3...
  26. twoski

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    Homework Statement Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 and y = 4 about the line y = 4 The Attempt at a Solution Since y=4 is parallel to the x-axis i assume i should be using the 'washer' method. The points of...
  27. O

    Cylindrical coordinates, finding volume of solid

    Homework Statement Find the volume of the solid that the cylinder r = acosθ cuts out of the sphere of radius a centered at the origin.Homework Equations Cylindrical coordinates: x = rcosθ, y = rsinθ, z=z, r2 = x2+y2, tanθ = y/xThe Attempt at a Solution So I know that the equation for the sphere...
  28. L

    Finding Volume of Solid: Isosceles Right Triangle Cross-Sections

    Hi, I'm still practicing how to find volume. 1. My problem is this: "Find the volume of the solid described below: The base of the solid is the disk x^2 + y^2 ≤ 4. The cross-sections by planes perpendicular to the y-axis between y=-2 and y=2 are isosceles right triangles with one leg in the...
  29. W

    Triple integral volume of solid (set up)

    Homework Statement I need help setting this integral up in spherical coordinates, the region above the xyplane, inside the sphere x^2+y^2+z^2=2 and outside the cylinder x^2+y^2=2 The Attempt at a Solution \int^{2\pi}_{0} \int^{\pi/2}_{\pi/4} \int^{\sqrt{2}}_{0}...
  30. W

    Volume of solid in region

    Homework Statement I want to convert this into polar and use double integral to find the volume of the solid in this region. I just need help setting this up region Q: x^2+y^2≤9, 0≤z≤4 I know this is a cylinder with a height of 4. I am just having trouble incorporating this height into the...
  31. M

    Finding the volume of solid

    Homework Statement the function is y = -x^2+6x -8 suppose a city is surrounded by a ring of mountains and these mountains can be illustrated by rotating the above function around the y-axis. Find the volume of the Earth that makes up these mountains. Suppose the city suffers from air...
  32. D

    Volume of Solid (Washer Method)

    Homework Statement Just need to verify that my working is correct ^^ Need to find the volume given by the region of the xy-plane that is bounded by the curves x = 0 and x = y − y2 . Rotated about y-axis Homework Equations The Attempt at a Solution I used the disk method. V = pi...
  33. D

    Volume of Solid (Washer Method)

    Homework Statement Question is: FInd the volume for area bound between y = x ^ (1/3) x = 4y About the x -axis I found the volumes from 0 to 8 using the washer method and then multiplied that by 2 since they intersect at y = 0, -2 and 2 x = 0, 8 and -8 Is that wrong? Cause my answer...
  34. C

    Find volume of solid elliptic paraboloid using polar coordinates

    Homework Statement a elliptic paraboloid is x^2/a^2+y^2/b^2<=(h-z)/h, 0<=z<=h. Its apex occurs at the point (0,0,h). Suppose a>=b. Calculate the volume of that part of the paraboloid that lies above the disc x^2+y^2<=b^2.:confused: 2. The attempt at a solution We normally do the...
  35. 2

    Find volume of solid rotated about X axis Washer method

    Homework Statement y=x^2+1 X in (0,1) Homework Equations formula needed integral (PI [f(x)]^2 DX in the F(x) just plug in the equation right? The Attempt at a Solution took anti deritive of the original problem and came out with PI[1/3x^3+X]^2 my answer is 5.58 am i on...
  36. I

    Finding Volume of Solid w/ Semicircular Cross-Sections

    x 0 0.5 1.0 1.5 2.0 2.5 3.0 f(x) 2 1.3 0.9 0.6 0.7 1.1 1.9 Find a formula for the volume V of the solid whose base is the region bounded by y = f(x), the x-axis, and the line x = 3 and its cross-sections perpendicular to the x-axis are semicircles.** So, I...
  37. T

    Finding maximum or minimum of volume of solid revolved about a region

    Question: For c>0, the graphs of y=(c^2)(x^2) and y=c bound a plane region. Revolve this region about the horizontal line y= -(1/c) to form a solid. For what value of c is the volume of this solid a maximal or minimal (Use calculus 1 techniques). First, I found the volume of this...
  38. E

    Volume of solid with given base and cross sections

    Homework Statement The base is the semicircle y = \sqrt{9-x2}(Square root of 9-x2.. i don't know why the formatting isn't showing up) where -3 <= x <= 3. The cross section perpendicular to the x-axis are squares. Homework Equations -3\int3 = A(y)dy A(y) = area of cross section...
  39. V

    Volume of Solid Revolution

    Hi all. I've just hit a block in the following question: [Find the volume of the solid...] "The region in the first quadrant bounded by the curve y = x^2, below by the x-axis, and on the right by the line x = 1, revolved around the axis x = -1." I've tried nearly 2 hours figuring the...
  40. R

    Volume of Revolution for Polynomial Bowl

    Homework Statement The question asked is to make a bowl out of polynomial equations rotated around the y axis. The bottom of the bowl has to have a maximum at the center and a minimum at some distance from the center. The equations I want to use are x^2+10, 1.3x^2 and -.7x^2 + 4. The...
  41. R

    Volume of Solid of Revolution Question

    Homework Statement y= -(x/6) + b, find the volume as this solid is rotated 360 degrees around the Y axis Homework Equations If I were given the interval at which I needed to find the volume and/or the value of B I could easily do this using the formula: [pi] Integrate: (R(y))2 dx...
  42. P

    Volume of solid with double integrals

    Homework Statement find volume of solid bounded by z=x, y=x, x+y=2 and z=0 The Attempt at a Solution first need to find domain. for x bounds, when y=0, x=0, when y = x, x+x=2 so x=1 therefore 0 < x < 1 for y bounds, x < y < 2-x now I am trying to work out what i integrate...
  43. B

    Volume of Solid using cross sections

    Homework Statement Find the volume of a solid whose base bounded by y=x+1 and y=x^2-1, with cross section of a square perpendicular to x-axis The Attempt at a Solution So i set up the problem like this delta volume= y^2*delta x, that being the area of the square where i get loss is...
  44. K

    Volume of solid bounded by 2 surfaces

    Just working on some practice problems. I missed a couple classes due to sickness and just need some extra help. If you could walk me through how to do these types of problems that would be amazing. Homework Statement Evaluate the volume of the solid bounded by the surfaces (x2 + y2)1/2 =...
  45. M

    Calc II finding volume of solid by rotating

    Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=x^2 x=y^2 about x= -1 Homework Equations Volume= Integral of A(y) dy where A(y)= (pi)(r)^2 The Attempt at a Solution My question is...
  46. S

    2nd semester calc question : calculating volume of solid of revolution

    Homework Statement The solid formed when the region bounded by y = x^2 and y = 2 - x^2 is revolved about the x-axis Homework Equations disc method with respect to x-axis the integral of : (pi * (f(x)^2 - g(x)^2)) The Attempt at a Solution When I square each function and...
  47. Y

    Triple integral to find Volume of Solid

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  48. D

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  49. L

    Volume of Solid Multivariable Calc

    Find the volume of the solid in the first octant of xyz space, bounded below by the coordinate axes and the unit circle and bounded about by z = 8xy A) 1/2 B) 1 C) 2 D) 4 E) 8 I know we need a double integral. The bound below should be the unit circle which would be x^2 + y^2 = 1. So...
  50. Z

    Volume of solid by cross-section question?

    Homework Statement I need to find the region bounded by these curves then find the volume of the solid generated by revolving this region about the x-axis. y= cscx, x= 1/4pi, x = 3/4pi, y=0 Homework Equations The Attempt at a Solution So I managed to sketch this region.. but I have trouble...
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