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Math Help - Evaluating a repeated integral

  1. #1
    Senior Member chella182's Avatar
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    Evaluating a repeated integral

    The question is:

    \int_{0}^{1}\int_{0}^{x}(5-3x-2y)dydx

    First I have to evaluate the repeated integral, which (I think) I can manage. But it then says "Then sketch the region of integration, reverse the order of integration, and evaluate the reverse integral."

    Now again, once I've understood the first bit of that question, I could probably evaluate the integral, but I don't know how to sketch the region of integration, or how to reverse the order of integration. Can anyone help?
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  2. #2
    MHF Contributor
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    Quote Originally Posted by chella182 View Post
    The question is:

    \int_{0}^{1}\int_{0}^{x}(5-3x-2y)dydx

    First I have to evaluate the repeated integral, which (I think) I can manage. But it then says "Then sketch the region of integration, reverse the order of integration, and evaluate the reverse integral."

    Now again, once I've understood the first bit of that question, I could probably evaluate the integral, but I don't know how to sketch the region of integration, or how to reverse the order of integration. Can anyone help?
    Hi

    Have a look to the sketch above

    To compute \int_{0}^{1}\int_{0}^{x}(5-3x-2y)dydx
    the area involved is the right-down triangle
    For a given x value (between 0 and 1), y can go from 0 to x (red line)

    To reverse the integral you can see that for a given y value (between 0 and 1), x can go from y to 1 (green line)

    Therefore integral is \int_{0}^{1}\int_{y}^{1}(5-3x-2y)dxdy

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  3. #3
    Senior Member chella182's Avatar
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    Thankyou I'm still not 100% with it, but my friend on my course has managed to tackle the question successfully, so hopefully he can shed some more light on the matter.
    The annoying thing is I was in the lecture for reversing the order as well! I don't know how I'm so rubbish at it.
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