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Math Help - Triple Integrals in Cylindrical coordinates

  1. #1
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    Triple Integrals in Cylindrical coordinates

    Find the volume of the region above the plane z=1/2 and below the sphere x^2+y^2+z^2=1 with a triple integral in cylindrical coordinates.
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  2. #2
    Member Danshader's Avatar
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    well from the question given this what i can think of:
    <br />
V = \int\int\int dV<br />

    <br />
V = \int^{1}_{-1} \int ^{\sqrt{1-x^2}} _ {-\sqrt{1-x^2}} \int ^{\sqrt{1-x^2 - y^2}} _{\frac{1}{2}} dzdydx ------- rectangular form

    rules of polar coordinates:
    r^2 = x^2 + y^2<br />
      x = r.cos\theta<br />
     y=r.sin\theta<br />

    <br />
V=\int^{2 \pi}_{0} \int ^{1} _ {0} \int ^{\sqrt{1-r^2}} _{\frac{1}{2}} r.dzdrd\theta--------------cylindrical form
    <br />
= \int^{2 \pi}_{0} \int ^{1} _ {0}  r.z |^{\sqrt{1-r^2}} _\frac{1}{2}.drd\theta<br />
    <br />
= \int^{2 \pi}_{0} \int ^{1} _ {0} r.(\sqrt{1-r^2} - \frac{1}{2}).drd\theta<br />
    <br />
= \int^{2 \pi}_{0} \int ^{1} _ {0} r\sqrt{1-r^2} - \frac{r}{2}.drd\theta <br />
    <br />
=\int^{2 \pi}_{0} -\frac{1}{3} {(1-r^2)}^{3/2}|^1 _0 - \frac{r^2}{4}|^1 _0.d\theta<br />
    <br />
=\int^{2 \pi}_{0} \frac{1}{12}.d\theta<br />
    <br />
=\frac{2\pi}{12}<br />
    <br />
=\frac{\pi}{6}<br />
    Last edited by Danshader; April 28th 2008 at 09:23 AM.
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