Results 1 to 4 of 4

Math Help - Volume of solid of revolution.

  1. #1
    Newbie
    Joined
    Dec 2010
    Posts
    3

    Volume of solid of revolution.

    1. Find the volume of the solid obtained by rotating the region bounded by the curve

    y^2 - y^3 - x = 0, x = 0


    about the line x = 0


    2. Integral of (cos x + xsinx)/x(x+cosx) dx
    Last edited by mr fantastic; December 16th 2010 at 11:59 AM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    Quote Originally Posted by devil10078 View Post
    1. Find the volume of the solid obtained by rotating the region bounded by the curve y^2 - y^3 - x = 0, x = 0 about the line x = 0
    The intersection points are (0,0) and (0,1) . Then,

    V=\pi\displaystyle\int_0^1(y^2-y^3)^2\;dy=\ldots

    Fernando Revilla
    Last edited by FernandoRevilla; December 15th 2010 at 11:46 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,553
    Thanks
    1424
    \displaystyle \frac{\cos{x} + x\sin{x}}{x(x+\cos{x})}=-\left(\frac{-\cos{x} - x\sin{x}}{x^2 + x\cos{x}}\right)

    \displaystyle = -\left(\frac{2x + \cos{x} - x\sin{x} - 2x- 2\cos{x} }{x^2 + x\cos{x}}\right)

    \displaystyle = -\left(\frac{2x + \cos{x} - x\sin{x}}{x^2 + x\cos{x}}\right) - \left(\frac{-2x - 2\cos{x}}{x^2 + x\cos{x}}\right)

    \displaystyle = -\left(\frac{2x + \cos{x} - x\sin{x}}{x^2 + x\cos{x}}\right) + \frac{2(x + \cos{x})}{x(x + \cos{x})}

    \displaystyle = \frac{2}{x} - \frac{2x + \cos{x} - x\sin{x}}{x^2 + x\cos{x}}.



    Therefore \displaystyle \int{\frac{\cos{x} + x\sin{x}}{x(x + \cos{x})}\,dx} = \int{\frac{2}{x} - \frac{2x + \cos{x} - x\sin{x}}{x^2 + x\cos{x}}\,dx}.

    To integrate the second term, make the substitution \displaystyle u = x^2 + x\cos{x}.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    13
    good call, but it's actually easier than that, since

    \begin{aligned}<br />
   \int{\frac{\cos x+x\sin x}{x(x+\cos x)}\,dx}&=\int{\frac{x+\cos x+x\sin x-x}{x(x+\cos x)}\,dx} \\ <br />
 & =\ln \left| x \right|-\int{\frac{1-\sin x}{x+\cos x}\,dx} \\ <br />
 & =\ln \left| x \right|-\ln \left| x+\cos x \right|+k \\ <br />
 & =\ln \left| \frac{x}{x+\cos x} \right|+k. <br />
\end{aligned}
    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: 12
    Last Post: July 3rd 2011, 05:33 AM
  2. volume of solid of the revolution
    Posted in the Calculus Forum
    Replies: 7
    Last Post: August 8th 2010, 08:01 AM
  3. volume of solid revolution around y=3
    Posted in the Calculus Forum
    Replies: 7
    Last Post: August 4th 2010, 07:02 AM
  4. volume of solid revolution
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 9th 2010, 08:54 AM
  5. volume of a solid of revolution?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 8th 2006, 03:06 AM

Search Tags


/mathhelpforum @mathhelpforum