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Math Help - Multiple integration using cylindrical cooridinates

  1. #1
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    Multiple integration using cylindrical cooridinates

    The question is : find the volume of the region bounded by the paraboloid z=1-4(x^2+y^2) and the xy plane. The answer to this question is 7pi/2. I have tried it but my answer comes out to be pi/8. The integration is correct i have checked with Mathematica, i think i have messed up the limits, maybe.

    I took the limits for z from 0 to 1-4r^2, for r from 0 to 1/2 and for theta from 0 to 2pi.


    Need help to figure out which one is incorrect.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Multiple integration using cylindrical cooridinates

    Quote Originally Posted by mrmaaza123 View Post
    I have tried it but my answer comes out to be pi/8.
    It is correct.

    \displaystyle\begin{aligned}V&=\displaystyle\iiint  _{T}dxdydz\\&=\displaystyle\int_0^{2\pi}d\theta \displaystyle\int_0^{1/2}dr\displaystyle\int_0^{1-4r^2}r\;dz\\&=2\pi\displaystyle\int_0^{1/2}(r-4r^3)\;dr\\&=2\pi\left[\frac{r^2}{2}-r^4\right]_0^{1/2}\\&=\frac{\pi}{8}\end{aligned}
    Thanks from mrmaaza123
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