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Can't interpret this question

  1. Oct 9, 2015 #1
    I don't think this goes in the homework section because I don't actually want help answering the question, I want to know what it means!

    Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple integral for the volume V using cylindrical coordinates. Include the limits of integration (three upper and three lower). Evaluate the integral to determine the volume V in terms of R.

    My main problem is when it asks about the cylinder x2 +y2 = 4R2. I'm nearly 100% sure that equation is not actually for a cylinder but for a circle! And I'm not entirely clear on whether I'm integrating two shapes, as in two volume integrals, or it's describing just one big shape.

    In the latter case, I still don't know where cylinders come into it.
     
  2. jcsd
  3. Oct 9, 2015 #2
    The equation is for a circle but you also have that ##z## varies between the ##xy##-plane and ##(x^2+3y^2)/R##. If you stack a lot of circles on top of each other you get a cylinder. So the first equation only describe one part of the cylinder while the third coordinate, ##z## is free too change value.
    So the equation is a circle if you were in a plane, If you were in 3d-space you have an infinite cylinder if you didn't have any restrictions on ##z##.
     
  4. Oct 9, 2015 #3
    I draw the shape of that cylinder with [itex] R=1 [/itex].
     

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  5. Oct 9, 2015 #4

    Mark44

    Staff: Mentor

    In three dimensions (which is implied by the statement that x, y, and z are coordinates), the equation (##x^2 + y^2 = 4R^2##) is a right circular cylinder. Since z does not appear in the equation, it is arbitrary.
     
  6. Oct 9, 2015 #5
    Arbitrary as opposed to zero?
     
  7. Oct 9, 2015 #6

    Mark44

    Staff: Mentor

    "Arbitrary" means "any value."
     
  8. Oct 9, 2015 #7
    I know. So arbitrary means it can take the value zero and others, as opposed to just zero, which is what I thought the equation meant. Shouldn't the z appear somewhere in the equation though? I feel like this is quite a basic concept I've misunderstood or missed! Oops!
     
  9. Oct 9, 2015 #8

    Mark44

    Staff: Mentor

    No, z doesn't have to appear in the equation. In the plane, the equation x = 2 is a vertical line. Here, y is not mentioned, and it is arbitrary, so every point in the plane with coordinates (2, y) is a point on this line. The situation is similar for your cylinder equation.
     
  10. Oct 9, 2015 #9
    Oh, I get it! Ok, that is a really important thing to know. Also know what the question is asking now! Thanks for your help :)
     
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