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    Hey,

    I'm stuck with this question:
    Calculate the volume enclosed by a cylinder with equation x^2+y^2=9 and the planes z=0 and x+z=5

    Can anybody give me some direction?
    Thank you
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    Should be able to do it from basic formulas and a bit of symmetry.
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    For your 3rd boundary condition do you mean z=5 rather than x+z=5 ?
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    (Original post by ve9jonny)
    For your 3rd boundary condition do you mean z=5 rather than x+z=5 ?
    I wish would make things a lot easier wouldn't it?
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    If yes then (pi x r^2 x h) = pi x 9 x 5 = 45pi which is simple and involves no integration.

    If no then it is not a cylinder as far I can tell
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    (Original post by ve9jonny)
    If yes then (pi x r^2 x h) = pi x 9 x 5 = 45pi which is simple and involves no integration.

    If no then it is not a cylinder as far I can tell
    No, it's not a cylinder.
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    The second plane slices the cylinder at an angle... first thing to do is work out the highest and lowest points of this slice .

    You'll find these points by using the line x + z = 5 along the diameter of the cylinder, y=0. You are able to choose the value since the plane is independent of y. So solve simultaneously for that line and the cylinder at y=0).

    Once you have the heights (z-coordinates) of these points, the volume is simply the cylinder to the lower point plus half the remaining portion (since by looking in the y direction it is clear you have half this last bit of cylinder: one triangle from a rectangle).

    Hope this makes things clearer.
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    (Original post by Visa Electron)
    No, it's not a cylinder.
    The volume bellow the z=f(x,y) and above the area of A on the z=0 plane
    \int \int_A f(x,y)\ dA and
    dA=dx \cdot dy
    Your A is the base of the cylinder that is x^2+y^2=9 \rightarrow y=\sqrt{9-x^2}
    As z+x=5, your f(x,y)=-x+5
    So
    V=\int^3_0 \int^{\sqrt{9-x^2}}_0 (-x+5)\ dy\ dx
    But it is better to calculate with polar coordinates
     x=r\cdot cos \phi and y=r\cdot sin \phi
    dA = r\cdot dr \cdot d \phi
    and limits of \phi 0 and 2\pi
 
 
 
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