Results 1 to 3 of 3

Math Help - Integration using transformations

  1. #1
    Junior Member SirOJ's Avatar
    Joined
    Dec 2009
    Posts
    45

    Integration using transformations

    Any ideas how I go about starting this problem. I cant find anything relevant in my notes..

    Use the transformation
    x = au, y = bv, z = cw to evaluate the integral

    integral integral integral(R) 1
    dV

    where
    R is the region bounded by the ellipsoid

    (x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1

    and hence calculate the volume of R.

    Any help would be much appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,347
    Thanks
    30
    Quote Originally Posted by SirOJ View Post
    Any ideas how I go about starting this problem. I cant find anything relevant in my notes..







    Use the transformation x = au, y = bv, z = cw to evaluate the integral

    integral integral integral(R) 1
    dV

    where


    R is the region bounded by the ellipsoid

    (x^2/a^2) + (y^2/b^2) + (z^2/c^2) = 1

    and hence calculate the volume of R.

    Any help would be much appreciated
    If V is the volume of the ellipsoid and U the volume of a sphere of radius 1 (which is what you get after your transformation) then

    <br />
\iiint_{V} 1 dV = \iiint_{U}1 \frac{\partial(x,y,z)}{\partial(u,v,w)}\, dU

    where \frac{\partial(x,y,z)}{\partial(u,v,w)} is the Jacobian of the transformation and equals abc. Thus,

    <br />
\iiint_{V} 1 dV = a b c \iiint_{U}1 \, dU = \frac{4}{3} \pi a b c.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member SirOJ's Avatar
    Joined
    Dec 2009
    Posts
    45
    Thanks alot for the reply... I'm not sure i really understand the answer to my last question. Is there any way you could step through the following question..?

    Use the variable transformation
    u = x y and v = x + y to evaluate the integral integral(S) sin(x y) cos(x + y) dA,

    where S is the square with vertices (0, 0), (pi/2, pi/2), (pi/2,-pi/2) and (pi, 0).

    Thank,
    SirOJ
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Transformations
    Posted in the Algebra Forum
    Replies: 6
    Last Post: March 6th 2011, 03:20 PM
  2. [SOLVED] Transformations
    Posted in the Pre-Calculus Forum
    Replies: 22
    Last Post: December 26th 2010, 03:52 PM
  3. Integration Transformations Proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 3rd 2009, 10:21 AM
  4. Transformations
    Posted in the Algebra Forum
    Replies: 2
    Last Post: September 13th 2009, 07:39 AM
  5. Transformations
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: October 27th 2007, 04:31 AM

Search Tags


/mathhelpforum @mathhelpforum