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Math Help - Short Triple Integration questions

  1. #1
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    Short Triple Integration questions

    Hello again,

    Just need a few short questions verified:

    1)
    In finding the centroid of the solid region of uniform density 'd' between the xy plane and the paraboloid z = 4 - (x^2 + y^2), the paraboloid opens down and we have a cicle about the xy plane, thus we can use cylindrical coordinates right? The big question is does this paraboloid only have symmetry with respect to all the three coordinate planes passing through the z axis? Therefore when finding the centroid (x,y,z) x=y=0 due to symmetry and we only need to find the moment with respect to z. Right?

    2)
    This is an improper integral question.

    I = double integral [ln(x)ln(y)/((x^2)*(y^2))] dA,
    D = [1, infinity] x [1, infinity], is it convergent or divergent?

    My question is can we simplify the integral by splitting it up with respect to x and y, that is, can we integrate ln(x)/x^2 with respect to x and multiply it by the integral ln(y)/y^2 with respect to y? This would be a lot simpler!

    The other question is: I know that ln(b) = + infinity, as b --> + infinity.
    Does (ln(b))/b approach zero as b --> + infinity?
    My confusion is if ln(b) = infinity, then we have for ln(b)/b = infinity/infinity = 1. However, when I plug actual numbers in for infinity, say b = 1000000, I get: ln(1000000)/1000000 = 13.81/1000000 = 1.38x10^-5 = 0. If this is try, I get a convergent improper integral with a value of 1.

    Thanks again.
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  2. #2
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    Quote Originally Posted by jcarlos

    2)
    This is an improper integral question.

    I = double integral [ln(x)ln(y)/((x^2)*(y^2))] dA,
    D = [1, infinity] x [1, infinity], is it convergent or divergent?
    \int_1^n \int_1^n \frac{\ln x \ln y}{x^2 y^2} dx dy
    I think you are having problem finding the iterated integral (partial antiderivatives).
    Here is how you do it.
    \int \frac{\ln u}{u^2} du
    Call
    v=\ln u and w'=u^{-2}
    Then,
    v'=1/u and w=-u^{-1}
    Thus,
    -\frac{\ln u}{u}+\int u^{-2} du
    Is,
    -\frac{\ln u}{u}-\frac{1}{u}+C
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  3. #3
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    Could somebody verify this for me please:

    I know that ln(b) = + infinity, as b --> + infinity; however,
    does (ln(b))/b approach zero as b --> + infinity?
    My confusion is if ln(b) = infinity, then we have for ln(b)/b = infinity/infinity = 1.
    However, when I plug actual numbers in for infinity, say b = 1000000, I get: ln(1000000)/1000000 = 13.81/1000000 = 1.38x10^-5 = 0. If this is try, I get a convergent improper integral with a value of 1.

    Also,
    In finding the centroid of the solid region of uniform density 'd' between the xy plane and the paraboloid z = 4 - (x^2 + y^2), the paraboloid opens down and we have a cicle about the xy plane, thus we can use cylindrical coordinates right? The big question is does this paraboloid only have symmetry with respect to all the three coordinate planes passing through the z axis? Therefore when finding the centroid (x,y,z) x=y=0 due to symmetry and we only need to find the moment with respect to z. Right?

    Thank you
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  4. #4
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    Quote Originally Posted by jcarlos
    Could somebody verify this for me please:

    I know that ln(b) = + infinity, as b --> + infinity; however,
    does (ln(b))/b approach zero as b --> + infinity?
    Yes.
    You have,
    \lim_{x\to \infty} \frac{\ln x}{x}
    This satisfites L-Hopital's rule.
    Thus,
    \lim_{x\to \infty} \frac{1}{x}=0
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  5. #5
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    Quote Originally Posted by jcarlos
    Could somebody verify this for me please:

    I know that ln(b) = + infinity, as b --> + infinity; however,
    does (ln(b))/b approach zero as b --> + infinity?
    My confusion is if ln(b) = infinity, then we have for ln(b)/b = infinity/infinity = 1.
    However, when I plug actual numbers in for infinity, say b = 1000000, I get: ln(1000000)/1000000 = 13.81/1000000 = 1.38x10^-5 = 0. If this is try, I get a convergent improper integral with a value of 1.
    You need to use L'hopital's rule to evaluate this.

    Alternatively if you know that \ln(x) goes to infinity more slowly
    than any positive power of x, then it is clear that your limit is 0.

    RonL
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  6. #6
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    A useful think to know that an exponential overtakes a polynomial and a polynomial overtakes the logarithmic.
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  7. #7
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    Thank you captain black and perfect hacker.

    I am also correct in the other question regarding the symmetry about all threee planes passing through the z axis for the cone and sphere question; therefore, when finding the moment of inertia, I only need to find the moment with respect to z as x = y = 0 due to symmetry.

    Thanks again
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