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Math Help - hard double integral

  1. #1
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    hard double integral

    Im stuck on this double integral, i cant see how it can be done!

    Outer limit 0 - pi/2 inner limit x - pi/2

    siny/y dydx


    also this one

    Outer limit 0 - 1 inner limit e^y - e

    x/lnx dxdy



    thanks
    Last edited by sterps; May 7th 2008 at 06:11 PM.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    <br />
\int_0^{\frac{\pi}{2}} \int_x^{\frac{\pi}{2}} y\cdot \sin y \, \mathrm{d}x \mathrm{d}y and \int_0^1 \int_{\exp y}^{\exp 1} x\cdot \ln x \, \mathrm{d}x\mathrm{d}y

    Are these the right integrals ?

    In both cases, you can first compute the inner integral...

     \int_0^{\frac{\pi}{2}} \int_x^{\frac{\pi}{2}} y\cdot \sin y \, \mathrm{d}x\mathrm{d}y= \int_0^{\frac{\pi}{2}} \int_x^{\frac{\pi}{2}} y\cdot \sin y \, \mathrm{d}y\mathrm{d}x=\int_0^{\frac{\pi}{2}}\left[\int_x^{\frac{\pi}{2}} y\cdot \sin y \, \mathrm{d}y\right]\mathrm{d}x

    Can you take it from here ?
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  3. #3
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    Krizalid's Avatar
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    For the second one, reverse integration order.
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  4. #4
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    thankyou for your responses, though i must not have not typed them out properly, the functions are actually quotients:

    siny / y dydx


    x / Lnx dxdy
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  5. #5
    Math Engineering Student
    Krizalid's Avatar
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    Well anyway in both cases reverse integration order.
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  6. #6
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    ill try that for the 2nd one, though ive done that for the first one

    outer limit: 0 to pi/2

    inner limit: 0 to y


    then should i use substitution to solve the integral?
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  7. #7
    Super Member flyingsquirrel's Avatar
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    then should i use substitution to solve the integral?
    No, there is no need to do this : reversing integration order brings simplifications.

    \int_0^{\frac{\pi}{2}}\int_x^{\frac{\pi}{2}}\frac{  \sin y}{y}\,\mathrm{d}x\mathrm{d}y=\int_0^{\frac{\pi}{2  }}\left[\int_0^y\frac{\sin y}{y}\,\mathrm{d}x\right]\mathrm{d}y=\int_0^{\frac{\pi}{2}}\left[\frac{\sin y}{y}\int_0^y\,\mathrm{d}x\right]\mathrm{d}y=\ldots
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  8. #8
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    thanks for that flying squirrel took straight after that, was good , i got an answer of -1, hope im right :S
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  9. #9
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by sterps View Post
    i got an answer of -1, hope im right :S
    I don't agree, the answer is 1. How did you integrate y\mapsto \sin y ?
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