Results 1 to 7 of 7

Math Help - Finding the volume of solids..

  1. #1
    Member
    Joined
    Nov 2012
    From
    Singapore
    Posts
    180

    Finding the volume of solids..

    Hi this is regarding volume of solids, it's the basics but I can't get the answers and it's frustrating me! Any help will be very much appreciated, really!

    1.) Find the volume of solid formed when y= x+\frac{1}{x} is rotated for 1\leq x \leq3
    I integrated (x+\frac{1}{x})^{2} from 1 to 3 and got 12pi units^3 as the answer, my answer after integration was pi( \frac{x^3}{3}+2x+\frac{1}{x})..I can't see any possible errors in my working, I must have integrated wrongly..

    2.) Find the volume of solid formed when y= \frac{x^3}{x^{2}+1} is rotated for xE[1,3]
    I integrated \frac{x^6}{x^{4}+1} from 1 to 3.
    I chose to use trig substitution, so I got x=\tan \theta
    I had changed x^{4}+1 to \sec^4 \theta since I thought \tan^2 \theta + 1 = \sec^2 \theta, then \tan^4 \theta + 1 = \sec^4 \theta..
    Simplifying, I ended up with \pi\tan^2 \theta\sin^4 \theta
    Also, I thought since \sin^2 \theta + \cos^2 \theta=1, then \sin^4 \theta + \cos^4 \theta=1
    so I got \pi\tan^2 \theta\(1-cos^4 \theta)..
    Separating into two integrals and changing tan to sec again, I ended up with
    \int^3_1\frac{3pi}{2}-\frac{cos\2\theta}{2}\dx ..
    Integrating I got
    \frac{1}{2}\pi\+\(3\theta+\frac{1}{2}\sin\2\theta), from 1 to 3..
    Are my steps correct so far?

    3.) A hemispherical bowl of radius 8cm is filled to a depth of 3cm. Find the volume of water in the bowl. The equation is \x^2+y^2=64
    I integrated from 0 to 3, the equation 64-y^2..

    Please assist me I really am at a loss as to why my answers are all wrong..

    Thank you very much!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,114
    Thanks
    988

    Re: Finding the volume of solids..

    (1) I assume the curve is rotated about the x-axis ...

    V = \pi \int_1^3 \left(x + \frac{1}{x}\right)^2 \, dx

    V = \pi \int_1^3 x^2 + 2 + \frac{1}{x^2} \, dx

    V = \pi \left[ \frac{x^3}{3} + 2x {\color{red}- \frac{1}{x}} \right]_1^3
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,114
    Thanks
    988

    Re: Finding the volume of solids..

    (2) note ...

    \left(\frac{x^3}{x^2+1}\right)^2 \ne \frac{x^6}{x^4+1}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2012
    From
    Singapore
    Posts
    180

    Re: Finding the volume of solids..

    For 2 is it x^6/(x^2+1)^2? I integrated with trig sub and got tan2x-x...please show me how you'd integrate, ibe a feeling im wrong..
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,114
    Thanks
    988

    Re: Finding the volume of solids..

    long division ...

    \frac{x^6}{x^4+2x^2+1} =

    x^2 - 2 + \frac{3x^2+2}{(x^2+1)^2} =

    x^2 - 2 + \frac{3x^2+3 - 1}{(x^2+1)^2} =

    x^2 - 2 + \frac{3(x^2+1)}{(x^2+1)^2} - \frac{1}{(x^2+1)^2} =

    x^2 - 2 + \frac{3}{x^2+1} - \frac{1}{(x^2+1)^2}

    integrating the first three terms is rather straightforward ... I would use the trig substitution x = \tan{t} for the last term.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Nov 2012
    From
    Singapore
    Posts
    180

    Re: Finding the volume of solids..

    Thank you!

    I wonder if there is something wrong with my method of long division, I got a different answer from yours..

    \frac{x^6}{(x^2+1)^3} = x^2 - \frac{2x^2-1}{x^4+2x^2+1}
    I used partial fractions to tackle the fraction..

    \frac{x^6}{(x^2+1)^3} = x^2 - \frac{2}{(x^2+1)}-\frac{3}{(x^2+1)^2}


    Thank you so much, really appreciate your help!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,114
    Thanks
    988

    Re: Finding the volume of solids..

    Quote Originally Posted by Tutu View Post
    I wonder if there is something wrong with my method of long division ...
    looks that way ...
    Attached Thumbnails Attached Thumbnails Finding the volume of solids..-longdivision.png  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 4th 2012, 03:42 PM
  2. Volume of solids of revolutions
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 14th 2010, 02:53 AM
  3. Volume of Solids
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2009, 07:26 PM
  4. Is This Right For Volume of Solids?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2009, 06:53 PM
  5. Volume of solids
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 22nd 2009, 06:17 PM

Search Tags


/mathhelpforum @mathhelpforum