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Math Help - double integral

  1. #1
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    double integral

    hi all!

    just started a course in multivariable calculus and i'm stuck at the problem of calculating the double integral dadb / (1 + a^2 + b^2)^2 over the domain b^2 < 2a.

    i guess i'm supposed to introduce polar coordinates, but i'm really unsure on the method, so i get a total mess.

    if someone could walk me through this i'd really appreciate it
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by flybyme View Post
    hi all!

    just started a course in multivariable calculus and i'm stuck at the problem of calculating the double integral dadb / (1 + a^2 + b^2)^2 over the domain b^2 < 2a.

    i guess i'm supposed to introduce polar coordinates, but i'm really unsure on the method, so i get a total mess.

    if someone could walk me through this i'd really appreciate it
    Are you sure about that domain. I might be missing something in your explanation of the problem (or I might have forgotten something from Calc 3), but shouldn't it be a closed domain? b^2 < 2a is open and infinite, and so the integration would be undefined.

    I'll let you work out the domain, I'll help setup the integration:

    DBL INT {over D} 1/(1 + a^2 + b^2) da db

    You are right in that we need to convert to polar coordinates (I'll let "O" represent "theta").
    Let b = rcosO
    Let a = rsinO
    Let da db = dA = r dr dO ... I hope you know how I got this

    DBL INT {over D} 1/(1 + r^2sin^2(O) + r^2cos^2(O)) r dr dO
    DBL INT {over D} 1/(1 + r^2) r dr dO

    Let 1 + r^2 = u <--> 2r dr = du --> r dr = 1/2 du

    DBL INT {over D} 1/(2u) du dO

    I can go no further because over the domain, D, r is a function of O. In order to integrate 1/(2u) du, we need to know the limits of r. Once you do know the limits, this integration becomes easy. After integrating r, you will be left with the integration of a function for O that is pretty straight forward, I think.

    INT {limits of O} f(O) dO, where f(O) is most likely some trig function.
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  3. #3
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    Quote Originally Posted by ecMathGeek View Post
    Are you sure about that domain. I might be missing something in your explanation of the problem (or I might have forgotten something from Calc 3), but shouldn't it be a closed domain? b^2 < 2a is open and infinite, and so the integration would be undefined.
    the domain is correct...

    I'll let you work out the domain, I'll help setup the integration:

    DBL INT {over D} 1/(1 + a^2 + b^2) da db
    you missed that it should be 1/(...)^2 da db

    INT {limits of O} f(O) dO, where f(O) is most likely some trig function.
    yeah, i think it gets to something like

    4 INT cos^2 O / (1 + cos^2 O)^2 dO, where O is between 0 and pi/2. but i have almost forgot how to solve these trig functions can't see how i can make this simpler, with some clever substitution
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