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Math Help - 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 h, inner radius r, and thickness \Delta r.

    This part is fairly simple-- dV=f'(r)*dr, assuming h is a constant. This yields 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 h is a constant, and thus doesn't account for the change in 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 \pi r^2 \Delta r. However, my book gives me an answer of \pi (\Delta r)^2 h. What am i doing wrong?
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    <br />
V(r)=\pi r^2 h<br />
    <br />
V'(r)=2 \pi r h<br />
    <br />
V''(r)=2 \pi h<br />

    <br />
\Delta V(r)=V'(r) \Delta r+ \frac {1}{2}V''(r) (\Delta r)^2<br />

    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 \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
    <br />
\Delta V(r)=V'(r) \Delta r+ \frac {1}{2}V''(r) (\Delta r)^2<br />

    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!

     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 h, inner radius r, and thickness \Delta r.

    This part is fairly simple-- dV=f'(r)*dr, assuming h is a constant. This yields 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 h is a constant, and thus doesn't account for the change in 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 \pi r^2 \Delta r. However, my book gives me an answer of \pi (\Delta r)^2 h. What am i doing wrong?
    edit- double post
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