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Math Help - Triple Integral

  1. #1
    Super Member Showcase_22's Avatar
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    Triple Integral

    Let \Omega be the three dimensional region defined by \Omega:=\{(x,y,z)| z\geq 0, x^2+y^2 \leq z^2 \leq 1-x^2-y^2 \}.

    a). Sketch the region \Omega
    b). Evaluate \int \int \int_{\Omega} dx \ dy \ dz

    Hint: It may be helpful to use spherical polar coordinates.
    a). For this part I got an ice cream cone shape. I tried drawing it on Matlab, but I kept getting red error messages!

    This ice cream cone has 0 \leq z \leq 1 and x^2+y^2= \left( \frac{1}{\sqrt{2}} \right)^2.

    I found this helpful for part b).!

    b). Is the required integral 2 \int_0^{\frac{1}{\sqrt{2}}} \int_0^z \int_{0}^y \ z \ dx \ dy \ dz+ \frac{1}{2} \int_0^{\frac{\pi}{2}} \int_0^{2  \pi} \int_0^{\frac{1}{\sqrt{2}}} \ r^3 sin (\phi) \ dr \ d \theta \ d \phi?

    Also, is there an easier way to write this integral???
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  2. #2
    MHF Contributor
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    Quote Originally Posted by Showcase_22 View Post
    a). For this part I got an ice cream cone shape. I tried drawing it on Matlab, but I kept getting red error messages!

    This ice cream cone has 0 \leq z \leq 1 and x^2+y^2= \left( \frac{1}{\sqrt{2}} \right)^2.

    I found this helpful for part b).!

    b). Is the required integral 2 \int_0^{\frac{1}{\sqrt{2}}} \int_0^z \int_{0}^y \ z \ dx \ dy \ dz+ \frac{1}{2} \int_0^{\frac{\pi}{2}} \int_0^{2  \pi} \int_0^{\frac{1}{\sqrt{2}}} \ r^3 sin (\phi) \ dr \ d \theta \ d \phi?

    Also, is there an easier way to write this integral???
    Hi

    OK for the ice cream

    Concerning the volume, I get something more simple than you

    \int_0^{\frac{\pi}{4}} \int_0^{2 \pi} \int_0^{1} \ r^2 sin (\phi) \ dr \ d \theta \ d \phi

    Anyway your calculation cannot be true because you are required to find the volume and integrating z \ dx \ dy \ dz and r^3 sin (\phi) \ dr \ d \theta \ d \phi do not lead to a volume
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  3. #3
    Super Member Showcase_22's Avatar
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    r^3 sin (\phi) \ dr \ d \theta \ d \phi do not lead to a volume
    This is probably a bit weird, but why...?

    I see what you've done for the integral and it is a lot better than what i've done.
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  4. #4
    MHF Contributor
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    Quote Originally Posted by Showcase_22 View Post
    This is probably a bit weird, but why...?
    You can see it as a matter of homogeneity (units)

    Volume are expressed by integrating :
    Cartesian : dx dy dz (which are all in meters, therefore volume is in cube meter)
    Cylindrical : dr r dtheta dz (dr, r and dz are in meters)
    Spherical : rē sin(phi) dr dtheta dphi (r and dr are in meters)
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