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Limits on an Integral of a semi-circle

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  • 28-05-2012 9:46pm
    #1
    Smythe
    Registered Users Posts: 434 ✭✭


    A question asks to calculate the integral over the region R given by:

    x^2 + y^2 <= 4
    0 <= y <= 2

    Which would be the upper half of a circle of radius 2 centred on the origin.

    The integral is done in the book I have and the limits of x are given as -2 to 2, which I can understand.

    Though the limits for y are given as: 0 to (4 - x^2)^0.5

    I can see that they have obtained this limit from rearranging the first part of the region R.

    BUT, why is the limit for y not 0 to 2. Or alternatively, if what they have done is correct, why is it not equally valid to state the limits for x are: 0 to (4 - y^2)^0.5


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    #2
    MathsManiac
    Registered Users Posts: 1,595 ✭✭✭MathsManiac


    You have to decide what order you're going to take the integral in.

    If you're going to take the integral with respect to x as the "outside" integral, then x can run from -2 to 2, but then you have to think about the values that y can take given a specified value of x, which is where they get 0 to (4 - x^2)^0.5.

    You could instead take them in the other order - taking the integral with respect to y on the "outside". In that case, y runs from 0 to 2, and you have to decide what x can be for a specified value of y. In this case, x could run from -(4 - y^2)^0.5 to (4 - y^2)^0.5 .

    You can think of it this way: do you want to "shade in" the region by drawing infinitely many vertical line segments, or by drawing infinitely many horizontal line segments.


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