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Math Help - Cylindrical co-ordinates, triple integration

  1. #1
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    Cylindrical co-ordinates, triple integration

    The integration i can do but i have no idea how you get the limits for to integrate by.

    I have to integrate a function where E (the limits) is the region enclosed by the planes z = 0 and z = x + y + 5 and by the cylinders X^2 + y^2 = 4 and x^2 + y^2 = 9.

    My take it is i first have to integrate with z between x + y + 5 and 0, but then i dont know how to get the next integrals with respect to x and y.

    Thanks for your help in advance.
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  2. #2
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    Quote Originally Posted by adam_leeds View Post
    The integration i can do but i have no idea how you get the limits for to integrate by.

    I have to integrate a function where E (the limits) is the region enclosed by the planes z = 0 and z = x + y + 5 and by the cylinders X^2 + y^2 = 4 and x^2 + y^2 = 9.

    My take it is i first have to integrate with z between x + y + 5 and 0, but then i dont know how to get the next integrals with respect to x and y.

    Thanks for your help in advance.
    Since you titled this "cylindrical coordinates, I do not understand why you are talking about "with respect to x and y". If you really wanted to use Cartesian coordinates, then you will need to divide the region between those two cylinders into non-overlapping portions where you can take y as a function of x or vice-versa.

    I presume that you recognize that x^2+y^2= 4 and x^2+ y^2= 9 are concentric cylinders of radii 2 and 3, repectively. when z= 0, the plane z= x+ y+ 5 becomes the line x+y= -5 which crosses the x and y axes at (0, -5) and (-5, 0). It's closest approach to the origin is where x= y: 2x= -5, x= y= -5/2. The distance from (-5/2, -5/2) to the origin is [tex]\sqrt{\frac{25}{4}+ \frac{25}{4}= \frac{5}{2}\sqrt{2}> 3. Whew! That means the top plane does not cut the lower plane inside the cylinders! That would have made the problem much harder!

    So: In cylindrical coordinates, \theta runs from 0 to 2\pi, r runs from 2 to 3, and z runs from 0 to z= x+ y+ 5= r cos(\theta)+ r sin(\theta)+ 5. That is:
    \int_{\theta= 0}^{2\pi}\int_{r= 2}^3\int_{z= 0}^{rcos(\theta)+ rsin(\theta)+ 5} f(r,\theta, z)r dzdrd\theta
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  3. #3
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    Hi guys. Guess I'm sort of amazed I can even plot these kinds of things:

    \mathop\int\int\int\limits_{\hspace{-25pt}\text{Red}} f(r,\theta,z)dV
    Attached Thumbnails Attached Thumbnails Cylindrical co-ordinates, triple integration-cylinderintegral.jpg  
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