Results 1 to 3 of 3

Thread: Switching the order of triple integrals

  1. #1
    Junior Member
    Joined
    Nov 2014
    From
    Home
    Posts
    70

    Switching the order of triple integrals

    Well this is just a nightmare and I got my first C on an exam in college after a year, going to finish calculus III with triple integrals and I can't figure this out at all

    the region is:

    Q: { (x,y,z) : x: [0,1] y: [0,x] and z: [0,3] }

    How do I find the boundaries of z? I can set up the integrals dzdxdy, dzdydx, I can also draw the region, I get the idea in 2d but I don't get how the bounds of integration for x are from 1 to y. How??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,795
    Thanks
    3035

    Re: Switching the order of triple integrals

    You are told that the "boundaries on z" are 0 and 3!

    Ignoring z, you are told that x goes from 0 to 1 and that, for each x, y goes from 0 to x. On a Cartesian coordinate system, draw vertical lines at x= 0 and x= 1 marking those boundaries. y= 0 is a third boundary and y= x, the fourth boundary goes from (0, 0) to (1, 1). Since y goes up from 0 to x, the area of integration is the triangle below y= x. To reverse the order of integration, observe that, over all, y goes from 0 to 1 and that, for every y, x goes from the line x= y to 1.

    $\displaystyle \int_{z= 0}^3\int_{x= 0}^1\int_{y= 0}^x f(x,y,z)dydxdz$
    $\displaystyle \int_{z= 0}^3\int_{y= 0}^1\int_{x= y}^1 f(x,y,z)dxdydz$
    $\displaystyle \int_{x= 0}^1\int_{z= 0}^3\int_{y= 0}^x f(x,y,z)dydzdx$
    $\displaystyle \int_{x= 0}^1\int_{y= 0}^x\int_{z= 0}^3 f(x,y,z)dzdydx$
    $\displaystyle \int_{y= 0}^1\int_{x= y}^1\int_{z= 0}^3 f(x,y,z)dzdxdy$
    $\displaystyle \int_{y= 0}^1\int_{z= 0}^3\int_{x= y}^1 f(x,y,z)dxdzdy$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2014
    From
    Home
    Posts
    70

    Re: Switching the order of triple integrals

    Thank you so much! That was extremely helpful.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Jun 5th 2015, 06:00 PM
  2. Triple Integrals
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 26th 2010, 02:29 AM
  3. Triple Integrals
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Oct 31st 2009, 08:42 AM
  4. triple integrals
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 4th 2009, 11:20 PM
  5. Triple Integrals
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Oct 6th 2008, 06:37 PM

/mathhelpforum @mathhelpforum