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Help finding double integrals

  1. Oct 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Finding the double integral of the following

    [tex]\int\int xy / (x^2+y^2+1)^1/2 dA[/tex]

    R = [(x,y): 0<=x<=1, 0<=y<=1]


    2. Relevant equations
    None


    3. The attempt at a solution

    ok I am having trouble integrating when I see the the quotient.

    what I have done is,

    [tex]\int\int xy(x^2+y^2+1)^-1/2 dy dx[/tex]

    I can't remember the step taken to ingrate this.

    I would add one to the exponent making it,

    (2)xy(x^2+y^2+1)^1/2 <---but I am missing something else. I would need to integrate each y right?

    so that would be

    [tex]\int x(x^2+ (1/3)y^3 +1 ) ^1/2 dx[/tex]

    and then I proceed integrating it again, this time for x... ???
     
    Last edited by a moderator: Oct 20, 2008
  2. jcsd
  3. Oct 19, 2008 #2

    Mark44

    Staff: Mentor

    The stuff you put in LaTex isn't formatting, so I'll have to go with what I think the problem is.

    You can change from the double integral over the region R to iterated integrals either by integrating with respect to y first and then x, or by integrating with respect to x first and then y. The region R is pretty simple, which makes matters simpler.

    Using the same order that you chose, the inner integral is from y = 0 to y = 1, and the integrand is xy/(x^2 + y^2 + 1)^(1/2), and you're integrating with respect to y.

    You can pull the factor of x outside the inner integral, since you're integrating w.r.t. y and x is constant on this strip.

    A straightforward substitution will work: u = x^2 + y^2 + 1, so du = 2ydy. Remember, x is not varying, so x can be considered to be a constant.

    Your integrand now looks like 1/2 u^(-1/2) du, and its antiderivative is 1/2 * 2 * u^(1/2) = u^(1/2).

    Undoing the substitution gives (x^2 + y^2 + 1)^(1/2).

    Evaluate the expression above at y = 1 and at y = 0, and subtract the second value from the first. This will leave you with a function of x alone.

    Now, integrate the resulting function w.r.t. x and evaluate its antiderivative at x = 1 and at x = 0, and subtract the second expression from the first.

    Don't forget that you pulled a factor of x out of the first integral; you'll need to incorporate this into the outer integrand.

    Hope that's enough to get you going.
     
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