1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solid of revolution question: verify that the volume of the cone is παβh/3

  1. Sep 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Consider a vertical cone of height h whose horizontal cross-section is an ellipse and whose base is the ellipse with major and minor semi-axes α and β. Verify that the volume of the cone is παβh/3.
    [ Hint: The area of an ellipse with major and minor semi-axes α and β is παβ. ]


    2. Relevant equations

    V = ∫A(y) dy (from c to d)
    V = ∫π(radius)² dy (from c to d)


    3. The attempt at a solution

    It says that the cone is upright, so I'm assuming it wants the cone rotated about the y-axis.
    V = ∫A(y) dy
    V = ∫π(radius)² dy

    Using similar triangles:
    x/y = r/h
    x = ry/h

    V = π∫(ry/h)² dy (the integral is now from 0 to h (c = 0, d = h))
    V = π∫(r²y²/h²) dy
    V = (πr²/h²)∫(y²) dy (since pi, r, and h are all constants)

    At this point I'm not sure where to go. Do I take the integral of y²? How do I incorporate α and β into this integral? (As a side note, I'm very new to these forums and if I've done anything wrong I apologize. I'm not used to writing out integrals on the computer and if the notation is not optimal I'm sorry!)
     
  2. jcsd
  3. Sep 15, 2010 #2
    Personally, I think you need to go out of your way to draw this thing, nicely. The volume via slices is:

    [tex]V=\int_0^h A(y)dy[/tex]

    with [itex]A(y)=\pi u(y) v(y)[/itex]

    and u and v are the minor and major axes as functions of y as you go up the cone starting from the base up to h.

    Since the axes at the base are [itex]\alpha,\beta[/itex], then as you said, using similar triangles, I get:

    [tex]u(y)=\frac{\alpha}{h}(h-y)[/tex]

    [tex]v(y)=\frac{\beta}{h}(h-y)[/tex]

    Alright, just integrate now.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Solid of revolution question: verify that the volume of the cone is παβh/3
Loading...