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Math Help - Evaluating Double Integrals

  1. #1
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    Evaluating Double Integrals

    Calculate the double integrals:

    I)
    IntInt R (x-y^2)dA

    where R is the region bounded by the curves y = x^2 and y = x^3

    II)
    IntInt D x^3 sin(y^3)dydx

    where D is the region bounded by the parabola y = x^2 and the two straight lines x = 0 and y = 1


    -------------

    I will be forever grateful for all the help I can get.
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  2. #2
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    Quote Originally Posted by Warrick2236 View Post
    Calculate the double integrals:

    I)
    IntInt R (x-y^2)dA

    where R is the region bounded by the curves y = x^2 and y = x^3

    II)
    IntInt D x^3 sin(y^3)dydx

    where D is the region bounded by the parabola y = x^2 and the two straight lines x = 0 and y = 1


    -------------

    I will be forever grateful for all the help I can get.
    Have you drawn the region of integration in each case?

    I) = \int_{x = 0}^{x = 1} \int_{y = x^3}^{y = x^2} x - y^2 \, dy \, dx.



    II) = \int_{x = 0}^{x = 1} x^3 \left( \int_{y = x^2}^{y = 1} \sin(y^3) \, dy \right) \, dx.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    [snip]
    II) = \int_{x = 0}^{x = 1} x^3 \left( \int_{y = x^2}^{y = 1} \sin(y^3) \, dy \right) \, dx.
    But since the y-integral is a tad tricky you might struggle with this order of integration

    So I'll let you consider how to reverse the order of integration ..... (You're again advised to first draw the region of integration).
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    (You're again advised to first draw the region of integration).
    Well actually one can even play with inequalities:

    In the dy\,dx order our region is bounded by 0\le x\le1,\,x^2\le y\le1, so 0\le x^2\le1 and from here 0\le x^2\le y\le1 and the limits for dx\,dy order will be 0\le y\le1,\,0\le x\le\sqrt y.

    Of course, this trick not always works, or may be hard sometimes if you only want to play with inequalities. In fact is useful for quick stuff.
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  5. #5
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    Thanks for showing the regions of the integration, but im still having some troubles solving both integration. And im not even sure if I get the correct answer.

    Any help would be highly appreciated!
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  6. #6
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    Quote Originally Posted by Warrick2236 View Post
    Thanks for showing the regions of the integration, but im still having some troubles solving both integration. And im not even sure if I get the correct answer.

    Any help would be highly appreciated!
    The hard work has been done. All you have to do is integrate.

    It would be best if you showed your working so that additional help can be targeted to your exact troubles.
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