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Math Help - Help with a multiple integral

  1. #1
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    Help with a multiple integral

    Hello,

    I need help integrating this integral. I tried doing it in wolfram alpha but I don't think it should be as complicated as the result it showed was.

    \iint\frac{xy^{2}}{1+y^{4}}dxdy

    Please just explain it to me or help me get started. I'm really not sure what to do.

    Thanks!
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Help with a multiple integral

    integrate x alone and y alone first begin with x since dx coming first

    can you integrate x I think you should know how to do that
    then y you have to integrate \frac{y^2}{1+y^4} can you ?
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  3. #3
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    Re: Help with a multiple integral

    Yeah I know how to get the integral with repect to x, but I don't know how to get the integral with respect to y from x^2 to 1
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  4. #4
    MHF Contributor Amer's Avatar
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    Re: Help with a multiple integral

    I am still trying to solve  \int \frac{y^2 }{y^4+1} \; dy

    but you can see here a wolfram solution it is messy click into show steps

    here
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    Re: Help with a multiple integral

    Quote Originally Posted by tubetess123 View Post
    Hello,

    I need help integrating this integral. I tried doing it in wolfram alpha but I don't think it should be as complicated as the result it showed was.

    \iint\frac{xy^{2}}{1+y^{4}}dxdy

    Please just explain it to me or help me get started. I'm really not sure what to do.

    Thanks!
    What are your bounds?
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  6. #6
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    Re: Help with a multiple integral

    The bounds are 0 \leq x \leq 1 and x^{2}\leq y \leq 1.

    Please show work. I get it down to the integral with just y and have no idea what to do. Might reversing the order of integration work? I really don't know.

    Thanks!
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  7. #7
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    Re: Help with a multiple integral

    Quote Originally Posted by tubetess123 View Post
    The bounds are 0 \leq x \leq 1 and x^{2}\leq y \leq 1.

    Please show work. I get it down to the integral with just y and have no idea what to do. Might reversing the order of integration work? I really don't know.

    Thanks!
    A better idea is for you to show some work. Reversing the order of integration might be a good idea, if the current order is causing problems. By looking at the integrand, it looks much easier to integrate with respect to x first...

    What will your new bounds be if you change the order?
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  8. #8
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    Re: Help with a multiple integral

    They should be  0 \leq x \leq \sqrt{y} and  0 \leq y \leq 1 .
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    Re: Help with a multiple integral

    Quote Originally Posted by tubetess123 View Post
    They should be  0 \leq x \leq \sqrt{y} and  0 \leq y \leq 1 .
    Correct, so now evaluate

    \displaystyle \begin{align*} \int_0^1{\int_0^{\sqrt{y}}{\frac{x\,y^2}{1 + y^4}\,dx}\,dy} \end{align*}
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  10. #10
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    Re: Help with a multiple integral

    But I thought we had to actually switch the order of integration? Doesn't that mean we have to switch dx and dy too? And switch which boundaries are on which integral sign?
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  11. #11
    MHF Contributor Amer's Avatar
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    Re: Help with a multiple integral

    \int_0^2 \int_0^{\sqrt{y}} \frac{xy}{1+y^4} \; dx\; dy = \int_{0}^{1} \frac{(\sqrt{y})^2y}{2(1+y^4)} - \frac{0(y)}{2(1+y^4)} \; dy = \int_{0}^{1} \frac{y^3}{2(1+y^4)} dy
    and this can be solved by sub u = 1+y^4
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  12. #12
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    Re: Help with a multiple integral

    Ok I got \frac{log(2)}{8}! Does that look right?
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  13. #13
    MHF Contributor Amer's Avatar
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    Re: Help with a multiple integral

    it is correct
    Wolfram
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  14. #14
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    Re: Help with a multiple integral

    Quote Originally Posted by tubetess123 View Post
    But I thought we had to actually switch the order of integration? Doesn't that mean we have to switch dx and dy too? And switch which boundaries are on which integral sign?
    The bounds you gave implied integration with respect to y first, so after you reverse the order, it implies an integration with respect to x first.
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  15. #15
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    Re: Help with a multiple integral

    So whether dx or dy is written first doesn't determine the order of integration?
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