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Math Help - Changing order of integration?

  1. #1
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    Changing order of integration?

    In The integral

     \int_1^{2} dx \int_{2-x}^{\sqrt{2x-x^2}} f(x,y)dy

    Change the order of integration.

    Can any1 tell me how to do this?
    i think y goes from 0 to 1 but i dont know how to do x.
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  2. #2
    Super Member
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    What's y doing? It's going from the lower curve y=2-x to the upper curve y=\sqrt{2x-x^2}. We then form the Riemann sum of that from x=1 to x=2. It helps to draw the curves. When we switch the order, integrating with respect to x. Now what is x doing? Well, x will be going across the same curves but in terms of x: from x=2-y to x=1-\sqrt{1-y^2} (complete the square). And then we form the Riemann sum of that and let y go from 0 to 1.

    \int_0^1\int_{2-y}^{1-\sqrt{1-y^2}} fdxdy
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