1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How do you set up an integral integrating two functions within a domain?

  1. May 28, 2008 #1
    How do you set up an integral integrating two functions within a domain??

    1. The problem statement, all variables and given/known data

    I have to integrate (2 + x + y) within the domain that is the area between 0 and 1 and (x+y<=1)
    I know how to integrate well, I think it's a double integral but I'm not really sure what the range is so not sure what to integrate from and to? Any help will be much appreciated. Thanks. I haven't done this kind of work since 1996 so it's been a while!
     
  2. jcsd
  3. May 28, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    If I read this correctly, you are to integrate f(x,y)= 2+ x+ y over the region bounded by x= 0, x= 1, y+ x= 1 and y= 0. (I added the last: without it or something similar the region is unbounded. Yes, that will be a double integral for two reasons: you are integrating a function of two variables and the region over which you are integrating is two dimensional.

    For any problem like this, you should draw a picture. Since the line x+ y= 1 goes throught both (1,0) and (0,1), that, together with x= 0 and y= 0, will give you a triangular region. You now need to decide in which order you want to integrate.

    If you decide to integrate with respect to y first, then with respect to x, you know that the limits of the "outer integral" (dx) must be numbers. Clearly x must range from 0 to 1 so the integral is from x= 0 to x= 1. Now, for each x, how must y range? draw a vertical line anywhere inside your triangle and look at it. The lower end is at the x-axis (y= 0) and the upper end is at x+ y= 1 or y= 1- x. The integral is
    [tex]\int_{x=0}^1\int_{y= 0}^{1-x} (2+ x+ y)dydx[/tex]
    Although most texts don't do it, I think it is a very good idea to write the "x= " and "y= " on the limits of integration like that.

    If you decide to integrate with respect to x first, then with respect to y, you know that the limits of the "outer integral" (dy) must show the total range of y: y must range from 0 to 1. For each y x ranges from x= 0 on the left to x= 1- y on the right:
    [tex]\int_{y=0}^1\int_{x= 0}^{1- y}(2+ x+ y)dxdy[/tex]

    Obviously those are exactly the same because of the symmetry of this problem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: How do you set up an integral integrating two functions within a domain?
Loading...