Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Volume of Revolution

  1. Jan 16, 2004 #1
    Included is my attempt at the following question. I get an answer of 10pi, whereas the right answer is (10pi)/3 from my text book. Here is the question:

    Rotate the triangle described by (-1,0),(0,1),(1,0) around the axis x=2 and calculate the volume of the solid.

    I basically changed the problem to the following and continued as it is the same

    Rotate the triangle described by (1,0),(2,1),(3,0) around the y-axis and calculate the volume of the solid

    Thanks for the help

    Attached Files:

  2. jcsd
  3. Jan 17, 2004 #2
    It is so small to understand the pic u quoted so i'm giving u my solution
  4. Jan 17, 2004 #3
  5. Jan 17, 2004 #4
    I don't quite understand it and I dont understand why mines wrong :(

    I'll keep trying to figure it out.
  6. Jan 17, 2004 #5


    User Avatar
    Staff Emeritus
    Science Advisor

    You may not have had it yet but this looks like an exercise in using Pappus' theorem: the volume of a solid of revolution is the area of a cross section times the circumference of the circle generated by the centroid of that cross section.

    In this case, the cross section is a triangle with base of length 2 and height 1: area= (1/2)(2)(1)= 1.
    The centroid (for a triangle only) is the "average" of the vertices:
    ((-1+0+1)//3,(0+1+0)/3)= (0, 1/3). The distance from (0, 1/3) to the line x= 2 is 2- 1/3= 5/3. The centroid "travels in" (generates) a circle of radius 5/3 and so circumference (10/3)pi.

    The volume of the figure is (1)(10/3)pi= (10/3)pi.
  7. Jan 17, 2004 #6
    Is this true for all kind of figure I never came across that theorem is there any link where i can go for reference
  8. Jan 18, 2004 #7


    User Avatar
    Staff Emeritus
    Science Advisor

    Pappus' theorem is true of any "solid of revolution". You should be able to find it in any calculus textbook (that includes multiple integrals.)
  9. Jan 18, 2004 #8
    I haven't learned that theorem so I don't really think I should use it.

    I'm trying to use horizontal rectangles for my area so I formulated the following integral which is how I was taught how to do questions like this.

    \int \pi r_o^2 - \pi r_i^2 dr



    and the limits of integration are from y=0 to y=1

    so we have this

    \int_0^1 \pi (-y+3)^2 - \pi (y+1)^2 dy

    it makes perfect sense to me but then once you work it out you get 10 pi. which is wrong

    So can someone point out what I did wrong and how to fix it so I dont do it again.

    Thanks very much.
    Last edited: Jan 18, 2004
  10. Jan 18, 2004 #9
    Here is the general formula

  11. Jan 18, 2004 #10
    does mine not make sense for some reason

    i am calculating the area of the circle when the farthest line is rotated and then subtracting the area of the circle when the closest line is rotated
  12. Jan 18, 2004 #11
    Yes it do makes sense dont you have gone to the previous post.

    Thats what u have to do and its general too

    Your way do make sense
  13. Jan 18, 2004 #12

    When you rotate a point about a line u get a circle


    When you rotate a line

    It would be somewhat like a truncated cone
  14. Jan 18, 2004 #13
    but i get the wrong answer so can you point out whats wrong with my formulation?

    keep in mind i moved the triangle to (1,0),(2,1),(3,0) because its the same problem correct?
  15. Jan 18, 2004 #14

    I'm rotating horizontal rectangles around the y-axis therefore each rectangle will make a circle
  16. Jan 18, 2004 #15
    Ok It would form rings as of saturn
  17. Jan 18, 2004 #16
    U can also do it analytically With no integration
  18. Jan 18, 2004 #17
    I still want to know whats wrong with this answer because it does not yield 10pi/3

    [tex]\int_0^1 \pi (-y+3)^2 - \pi (y+1)^2 dy[/tex]
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Volume of Revolution
  1. Volumes by revolutions (Replies: 2)

  2. Volume of revolution (Replies: 7)