1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Volumes by Cylindrical Shells

  1. Nov 7, 2004 #1


    User Avatar

    Hi, I was hoping someone could check my work on a few problems and get me started on a few others. It involves definite integration, so I'm gonna use (a,b)S as an integration symbol and P for pi.

    These are the ones I need checked:
    1. Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the x-axis.
    x=2y, y=2, y=3, x=0
    2P*(2,3)Sy(2y)dy = 4P*(2,3)S(y^2)dy = 4P/3*[y^3](2,3) = 76P/3

    2. Use cylindrical shells to find the volume of the solid that is generated when the region that is enclosed by y=1/x^3, x=1, x=2, y=0 is revolved about the line x=-1
    2P*(1,2)S(x/(x-1)^3) = -2P*[(1-2x)/(2(x-1)^2)](1,2) =
    I'm stuck here, because putting 1 into the equation puts a zero in the denominator.

    3. (a) Find the volume V of the solid generated when the region bounded by y=1/(1+x^4), y=0, x=1, and x=b (b>1) is revolved about the y-axis.
    (b) Find lim(b->+infinity) V
    (a) 2P*(1,b)S(x/(1+x^4)) = 2P*[(x^2)/2 - 1/(2x^2)](1,b) = 2P(.5b^2 - 1/(2b^2))
    (b) Infinity

    4. The base of a certain solid is the region enclosed by y = x^.5, y=0, and x=4. Every cross section pependicular to the x-axis is a semicircle with its diameter across the base. Find the volume of the solid.
    P/16*(0,4)Sxdx = .5P

    These are the ones where I don't even know where to start:
    5. The region enclosed between the curve y^2=kx and the line x=.25k is revolved about the line x=.5k. Use cylindrical shells to find the volume of the resulting solid. (Assume k>0)

    6. Use cylindrical shells to find the volume of the torus obtained by revolving the circle x^2 + y^2 = a^2 about the line x=b, where b>a>0. [Hint: It may help in the integration to think of an integral as an area.]

    Much thanks to anyone who can give me any help. I really appreciate it.
  2. jcsd
  3. Nov 7, 2004 #2

    Math Is Hard

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Hi Bri,
    The first one looks good, but that's all I have had time to look at. I am an old lady and very slow! :smile: And it's been a while since i've done any shelling!
    I am hoping this post will inspire some others to jump in.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook