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Double integral problem, conceptual help.

  1. Mar 16, 2015 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    Find the volume: Prism formed by x+z=1, x-z=1, y=2, y=-2, and the yz-plane.

    2. Relevant equations


    3. The attempt at a solution
    Okay, so I sketched the drawing and I found that I could take the upper region of the xy-plane with respects to x and z and a triangle was formed.
    The bounds were from 0 to 1 with respects to x and from 0 to z=1-x with respects to z.
    So i integrated dzdx and got the answer of 1/2 for the triangle.

    I then multiplied the triangle area by 2, since we have to take the volume of above the xy-plane and below the xy-plane which due to symmetry, I assume this works.

    I then multiplied by 4 as y=-2 and y=2 distance results in 4 units.

    So my answer came to be "4".

    This seems a bit too...easy, however, I can't find anything conceptually wrong with my procedure.

    Did I perform this operation correct?
     
  2. jcsd
  3. Mar 16, 2015 #2
    If y = f(x, z) ≡ 2 & by only calculating the part of the prism in one octant then multiplying by 4 (symmetry) I got ##4\int_{0}^{1}\int_{0}^{1-x} f(x, z) dz\,dx = 4\int_{0}^{1}\int_{0}^{1-x} 2\,dz\,dx = 4##. That's the same as I got by just doing (length)x(width)x(height)/2 so it seems right....
     
  4. Mar 17, 2015 #3

    RJLiberator

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    Oh wow, that is awesome. It made sense to me in my head, but it seemed like I was missing something. I'm glad we got the same answer. Thank you for checking.
     
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