Results 1 to 4 of 4

Math Help - [SOLVED] Volume of Solid Of Revolution....y=sqrt(x) and y=x, around x=5

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    53

    [SOLVED] Volume of Solid Of Revolution....y=sqrt(x) and y=x, around x=5

    Hello,

    I have been trying to solve the following problem:

    Consider the solid obtained by rotating the region bounded by the given curves about the line x = 5. The curves are: y=\sqrt{x} and y=x. Find the volume V of this solid.

    Facts: The only points of interesction are at (0,0) and (1,1).

    Attempted Solution:

    We need to use the washer method. However, since the region is being rotated around the line x=5, the radius of each circle will be r = r + 5. We also need to integrate from 0 to 1 along y. Therefore, my solution looked something like this:

    V = pi \int (y^2+5)^2dy - pi \int (y+5)^2dy

    expanding....

    V = pi \int y^4+10y^2+25dy - pi \int y^2+10y+25dy

    simplify...

    V= pi \int y^4+9y^2-10y dy

    integrate...

    V = pi(\frac{y^5}{5} + 3y^3 - 5y^2)

    V = pi(\frac{1}{5} + 1 - \frac{5}{2})

    V = \frac{-21}{10}pi



    The problem with my solution is that volume can not be negative. Can some one please explain what is wrong with my solution?

    Thanks,
    Last edited by calc101; May 28th 2009 at 11:47 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,825
    Thanks
    714
    Hello, calc101!

    Did you make a sketch?
    Your radii are wrong . . .


    Consider the solid obtained by rotating the region bounded by the given curves
    about the line x = 5. .The curves are: y\:=\:\sqrt{x}\text{ and }y\:=\:x.
    Find the volume V of this solid.

    Facts: The only points of interesction are at (0,0) and (1,1).

    Attempted Solution:

    We need to use the washer method.
    However, since the region is being rotated around the line x=5,
    the radius of each circle will be r = r + 5. . . . . no
    We also need to integrate from 0 to 1 along y.
    Code:
          |                 :
          |                 *
          |      ...*       :
          |   *:::/         :
          | *:::/           :
          |*::/             :
          |:/               :
        - * - - - - - - - - + - -
          |                 5

    Our functions are: . x \,=\,y^2,\;x \,=\,y

    The outer radius is: . r_1 \:=\:5-y^2
    The inner radius is: . r_2 \:=\:5-y

    Hence: . V \;=\;\pi\int^1_0(5-y^2)^2\,dy - \pi\int^1_0(5-y)^2\,dy

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303
    You integrated wrong. Your set up needs to be changed anyways...

    Try....
    <br />
V \;=\;\pi\int^1_0y^4-11y^2+10y\,dy<br />

    Oh man, didn't see the guy above me already worked it out. His work is the same as mine though. You just messed up the radius calculation...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    May 2009
    Posts
    53
    gosh, i feel stupid....why of course....the radii is five points from the origin minus the change produced by the functions...

    I did sketch it, but it didn't quite click...

    thanks a lot guys...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Volume of solid of revolution
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 9th 2010, 10:57 PM
  2. Volume of solid of revolution.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 31st 2009, 08:33 PM
  3. Volume of solid of revolution
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 7th 2009, 05:13 AM
  4. Volume of Solid of Revolution
    Posted in the Calculus Forum
    Replies: 0
    Last Post: June 8th 2009, 10:52 AM
  5. [SOLVED] [SOLVED] Volume of a Solid Revolution
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 13th 2006, 05:33 PM

Search Tags


/mathhelpforum @mathhelpforum