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Changing the order of integration

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  • 24-04-2012 9:52pm
    #1
    jeepers101
    Registered Users Posts: 707 ✭✭✭


    I have no idea how to write the question so I've scanned and attached my attempt.

    The original integral, as I understand it is integrating from y=sinx to y=3sinx from o to π. (Vertical arrow)

    Now I want to integrate in the direction of the horizontal arrow.

    My problem is that in my graph, above y=1, I cannot integrate from x=arcsin(y/3) to x=arcsiny. I am just integrating from x=arcsin(y/3) to itself.

    I suspect I need to break the graph into more than one section and present the solution as the sum of these sections but I don't know how to do this.

    Any help is much appreciated.


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    #2
    jeepers101
    Registered Users Posts: 707 ✭✭✭jeepers101


    I think I have solved it.

    I have broken the graph into four sections. Two of the sections are to the right of x = π/2 and the other two are to the left. Because of the symmetry I can just concentrate on the left hand side and double the answer.

    So now the first section I will integrate from x=arcsin(y/3) to x=arcsiny between 0 and 1.

    For the second section I will integrate from x=arcsin(y/3) to x=π/2 between 1 and 3.

    Now as I said already, double the sum of these integrals.

    * π is pi.


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