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Math Help - Volume of a cylinder restriced by planes

  1. #1
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    Volume of a cylinder restriced by planes

    Find the area of a cylinder x^2+y^2=9 , the planes y+z=1 and z=1

    This is how I set it up: ∫ ∫ ∫ (x^2+y^2) dz dy dx

    => z bounded by 1 < z < 5-y

    ∫ ∫ ∫ (x^2+y^2) dz dy dx = ∫ ∫ (4-y)(x^2+y^2) dy dx

    => using x = r cos T, y = r sin T, I get:

    ∫ ∫ (4-rsin T)(r^2)(r) dr dT where 0 < r < 3, 0 < T < 2pi.

    I try to calculate this and I get the wrong answer. Did I set this up incorrectly? I was hoping it was just a computational error, but I have double checked it and still get the wrong answer, which is supposed to be 36pi.

    Thanks!
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  2. #2
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    Thanks for the help, but I have another question ><

    Doesn't the fact that the cylinder is bounded by z=5-y matter? I don't see how that is built into the integral you wrote up there
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  3. #3
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    Quote Originally Posted by crabchef View Post
    Find the area of a cylinder x^2+y^2=9 , the planes y+z=1 and z=1

    This is how I set it up: ∫ ∫ ∫ (x^2+y^2) dz dy dx

    => z bounded by 1 < z < 5-y

    ∫ ∫ ∫ (x^2+y^2) dz dy dx = ∫ ∫ (4-y)(x^2+y^2) dy dx

    => using x = r cos T, y = r sin T, I get:

    ∫ ∫ (4-rsin T)(r^2)(r) dr dT where 0 < r < 3, 0 < T < 2pi.

    I try to calculate this and I get the wrong answer. Did I set this up incorrectly? I was hoping it was just a computational error, but I have double checked it and still get the wrong answer, which is supposed to be 36pi.

    Thanks!
    The correct set up for the integrals is:

    V = \int \int_{R_{xy}} \int_{z = 1 - y}^{z=1} dx\, dy, dz

    where R_{xy} is the upper half of the circle x^2 + y^2 = 9 in the xy-plane.

    The surfaces defining the volume get built into the integral terminal, NOT the integrand (which is the mistake you made).
    Last edited by mr fantastic; July 16th 2008 at 06:56 PM. Reason: Corrected the region in the xy-plane.
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  4. #4
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    Quote Originally Posted by crabchef View Post
    Thanks for the help, but I have another question ><

    Doesn't the fact that the cylinder is bounded by z=5-y matter? I don't see how that is built into the integral you wrote up there
    You've gave the plane y+z=1 in your original question. So which is it: y + z = 1 or y + z = 5?
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  5. #5
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    ah sorry! it was in fact y + z =5
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  6. #6
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    Quote Originally Posted by crabchef View Post
    ah sorry! it was in fact y + z =5
    Draw a large and well labelled diagram (I found a side view in the zy-plane extremely helpful) and you easily see that V = V1 + V2 where:

    V1 is the volume of a cylinder of height 1 and radius 3: V_1 = 9 \pi.

    V2 is half the volume of a cylinder of height 6 and radius 3: V_2 = \frac{1}{2} \pi (3)^2 (6) = 27 \pi.


    Alternatively:

    V = \int \int_{R_{xy}}(5 - y) - 1 \, dx \, dy = \int \int_{R_{xy}}4 - y \, dx \, dy

    where R_{xy} is the region of the xy-plane defined by the circle x^2 + y^2 = 9.
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  7. #7
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    thanks for the help. It helps a bit, but I'm still having a lot of trouble in this class (just your basic multi var course)! ugh it's frustrating...
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