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Math Help - Triple Integral Limits

  1. #1
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    Triple Integral Limits

    I don't understand how the limits of integration have been obtained in the following worked example:




    So, how did they get the limits \int^{2 \pi}_0 \int^1_0 \int^1_r?

    I know that for the double integral they've used polar coordinates. The \int^{2 \pi}_0 \int^1_0 represents the unit circle. I tried to sketch the region but I'm not sure how the solid is supposed to look like...

    Especially I don't know how they got limits " \int^1_r"! Does \sqrt{x^2 + y^2} represent a hemisphere?

    I really appreciate it if anyone could explain these to me.
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  2. #2
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    Quote Originally Posted by demode View Post
    [snip]

    Especially I don't know how they got limits " \int^1_r"! Does \sqrt{x^2 + y^2} represent a hemisphere?
    Yes it does. The whole region is, in fact, a cylinder with a hemisphere removed, so that there are sharp edges along the unit circle in the xy plane and there is nothing on the z axis.

    Actually, I just figured out a great way to think of this without pictures!

    If your only concern is the z bounds, they're actually spelled out in that first condition. For all (x,y) z goes between 1 and sqrt(r^2) = r.
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  3. #3
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by Turiski View Post
    Yes it does. The whole region is, in fact, a cylinder with a hemisphere removed, so that there are sharp edges along the unit circle in the xy plane and there is nothing on the z axis.

    Actually, I just figured out a great way to think of this without pictures!

    If your only concern is the z bounds, they're actually spelled out in that first condition. For all (x,y) z goes between 1 and sqrt(r^2) = r.
     \sqrt{ x^2 + y^2 } does NOT represent a hemisphere. It represents a cone.

    The reason for the bounds of

     \int_r^1 dz

    is because they are given by  \sqrt{ x^2 + y^2 } \le z \le 1

    In cylindrical co-ordinates  x^2 + y^2 = r^2 so

     \sqrt{ x^2 + y^2 } \le z \le 1 \to r  \le z \le 1
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