Results 1 to 2 of 2

Math Help - triple integral volume problem

  1. #1
    Member
    Joined
    Oct 2007
    Posts
    159

    triple integral volume problem

    I need to find the volume of region bound by z = x + y z = 10 and the planes x= 0 and y = 0

    I set up an integral from 0 to 10 then from 0 to 10 - x and from 10 to x + y dzdydx with the function being one dzdydx

    I get an answer 5 times the amount I should. Could someone please tell me if I miscalculated the bounds on one of my integrals??? Thanks so much for the assistance, Frostking
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2008
    Posts
    319
    Here, z=x+y is a plane sloping diagonally upward along the x- and y-axes that forms a tetrahedron with the planes x,\,y=0 and z=10. We can always use the formula for the volume of a cone:

    V=\frac{1}{3}Bh,

    where B is the base area, here 50. You are correct that the integral to evaluate is

    \int_0^{10}\int_0^{10\,-\,x}\int_{x\,+\,y}^{10}dz\,dy\,dx.

    Carrying out two steps, we find that this equals

    \begin{aligned}<br />
\int_0^{10}\int_0^{10\,-\,x}(10-x-y)\,dy\,dx &=\int_0^{10}\left[10y-xy-\frac{y^2}{2}\right]_0^{10\,-\,x}\,dx\\<br />
&=\int_0^{10}\left(10(10-x)-x(10-x)-\frac{1}{2}(10-x)^2\right)\,dx\\<br />
&=\int_0^{10}\left(100-10x-10x+x^2-\frac{x^2}{2}+10x-50\right)\,dx.<br />
\end{aligned}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. volume, triple integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 17th 2010, 09:09 PM
  2. triple integral - volume
    Posted in the Calculus Forum
    Replies: 6
    Last Post: November 29th 2009, 09:03 AM
  3. Volume problem (triple integral)
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 4th 2009, 01:59 PM
  4. Triple integral - volume
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 14th 2008, 11:39 PM
  5. triple integral volume problem
    Posted in the Calculus Forum
    Replies: 8
    Last Post: July 24th 2006, 07:23 PM

Search Tags


/mathhelpforum @mathhelpforum