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Math Help - Triple integral

  1. #1
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    Triple integral

    Could someone please help me with this problem?

    Calculate the integral
    I = \int {\int\limits_D { \int {xdxdydz} } }

    where D is the region in R3 given by the inequalities x^2+y^2+z^2<=1 and -x<=y<=\sqrt{3}x

    How do you deal with that last inequality?
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  2. #2
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    Quote Originally Posted by kloda View Post
    Could someone please help me with this problem?

    Calculate the integral
    I = \int {\int\limits_D { \int {xdxdydz} } }

    where D is the region in R3 given by the inequalities x^2+y^2+z^2<=1 and -x<=y<=\sqrt{3}x

    How do you deal with that last inequality?
    The first part is a\the inside of a sphere of radius 1

    the region in the xy plane is bounded by two lines see attached graph

    Triple integral-capture.jpg

    I would change to cylindrical coordinates from here.
    If you haven't covered them yet this is it in rectangular
     <br />
\int_{0}^{1}\int_{-x}^{\sqrt{3}x}\int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}}xdzdydx<br />

    cylindrical coordinates give

    \int_{-\frac{\pi}{4}}^{\frac{\pi}{3}}\int_{0}^{1}\int_{-\sqrt{1-r^2}}^{\sqrt{1-r^2}}(r\cos(\theta))rdzdrd\theta

    I hope this helps.

    Good luck.
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  3. #3
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    Ah, thank you so much. It's so obvious now.
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