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Thread: Volume of a cylindrical shell

  1. #1
    Member Chokfull's Avatar
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    Volume of a cylindrical shell

    (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height $\displaystyle h$, inner radius $\displaystyle r$, and thickness $\displaystyle \Delta r$.

    This part is fairly simple--$\displaystyle dV=f'(r)*dr$, assuming $\displaystyle h$ is a constant. This yields $\displaystyle dV=2\pi rh\Delta r$.

    (b) What is the error involved in using the formula from part (a)?

    This is where I'm stuck. The formula in part (a) assumes $\displaystyle h$ is a constant, and thus doesn't account for the change in $\displaystyle h$, even though the thickness does affect the top of the shell. Seeing this, I would assume the error to be in the volume of the "lid", or $\displaystyle \pi r^2 \Delta r$. However, my book gives me an answer of $\displaystyle \pi (\Delta r)^2 h$. What am i doing wrong?
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  2. #2
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    $\displaystyle
    V(r)=\pi r^2 h
    $
    $\displaystyle
    V'(r)=2 \pi r h
    $
    $\displaystyle
    V''(r)=2 \pi h
    $

    $\displaystyle
    \Delta V(r)=V'(r) \Delta r+ \frac {1}{2}V''(r) (\Delta r)^2
    $

    The second term may be considered to be an error term for the first one.
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  3. #3
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    Also, the real volume is $\displaystyle \pi(r+\Delta r)^2h-\pi r^2h=2\pi r\Delta rh+\pi(\Delta r)^2h$.
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    Member Chokfull's Avatar
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    Quote Originally Posted by zzzoak View Post
    $\displaystyle
    \Delta V(r)=V'(r) \Delta r+ \frac {1}{2}V''(r) (\Delta r)^2
    $

    The second term may be considered to be an error term for the first one.
    Where did you get this formula?
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    Member Chokfull's Avatar
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    Well, I'm not sure this is the correct way I'm supposed to find the answer, since I haven't learned about Taylor series yet. Is there another way?
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  7. #7
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    What about my suggestion above?
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  8. #8
    Member Chokfull's Avatar
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    Well, yes, geometrically that is the correct formula. The point here, though, is to use differentials. There is supposed to be room for error in the final equation, so I assume I'm supposed to use a linear approximation somehow.
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    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by Chokfull View Post
    Where did you get this formula?
    This also comes from what we learn in high school physics!

    $\displaystyle D = vt + \frac{1}{2}at^2 $ where D is distance, V is velocity, t is time and a is acceleration.
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  10. #10
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by Chokfull View Post
    (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height $\displaystyle h$, inner radius $\displaystyle r$, and thickness $\displaystyle \Delta r$.

    This part is fairly simple--$\displaystyle dV=f'(r)*dr$, assuming $\displaystyle h$ is a constant. This yields $\displaystyle dV=2\pi rh\Delta r$.

    (b) What is the error involved in using the formula from part (a)?

    This is where I'm stuck. The formula in part (a) assumes $\displaystyle h$ is a constant, and thus doesn't account for the change in $\displaystyle h$, even though the thickness does affect the top of the shell. Seeing this, I would assume the error to be in the volume of the "lid", or $\displaystyle \pi r^2 \Delta r$. However, my book gives me an answer of $\displaystyle \pi (\Delta r)^2 h$. What am i doing wrong?
    edit- double post
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