Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By hollywood

Math Help - Volume of solid, varying angle

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    40

    Volume of solid, varying angle

    Hey, had another I could use a hand with. My guess is that I'm reading the task wrong.

    Problem:
    "Compute the volume of a solid that has height h if its horizontal cross section at any height y above its base is a circular sector having radius a and an angle t = 2*pi*(1-(y/h))."

    Meandering attempts at solution:
    The way I interpret it, we have a rather strange solid, which pinches like the middle of an hour glass at y = h/4 and y = 3h/4, and has the full circular area perpendicular to the height at
    y = 0, y = h/2 and y = h.

    Well, as far as I am aware, you can generally find the volume of any solid whose cross sectional area A(y) is known as a function of the height y by taking the integral of
    A(y)dy, in this case between the limits y = 0 and y = h.

    The way I want to read it, the cross sectional area should be
    A(y) = pi*a2*cos(t)

    I then try to simply solve the aforementioned integral, substituting u = t, but then I seem to run into a wall and I'm not sure if it's just me bungling the integral calculation or if I've misunderstood the wording of the task. According to the key the correct answer should be pi*a2*h/2 cubic units.
    Last edited by Scurmicurv; April 10th 2013 at 06:04 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: Volume of solid, varying angle

    The sector is \frac{t}{2\pi} of a whole circle, so its area is \frac{t}{2\pi}\pi a^2=\frac{1}{2}ta^2. So you substitute the equation for t and integrate from 0 to h dy. The answer the key gives is correct.

    - Hollywood
    Thanks from Scurmicurv
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2010
    Posts
    40

    Re: Volume of solid, varying angle

    Well, sure, when you put it like that it's trivial. Weird how you can sometimes stare yourself blind at stuff like this. Anyhows, thanks a lot for the assist!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Volume of solid
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 5th 2011, 04:13 AM
  2. Solid Angle
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: September 1st 2011, 11:45 AM
  3. Volume of solid
    Posted in the Calculus Forum
    Replies: 7
    Last Post: February 11th 2011, 02:12 AM
  4. Replies: 2
    Last Post: July 6th 2010, 06:33 PM
  5. volume of a solid
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 6th 2009, 04:29 PM

Search Tags


/mathhelpforum @mathhelpforum