# Thread: Volume of a cylindrical shell

1. ## 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?

2. $
V(r)=\pi r^2 h
$

$
V'(r)=2 \pi r h
$

$
V''(r)=2 \pi h
$

$
\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.

3. Also, the real volume is $\pi(r+\Delta r)^2h-\pi r^2h=2\pi r\Delta rh+\pi(\Delta r)^2h$.

4. Originally Posted by zzzoak
$
\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?

5. 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?

6. What about my suggestion above?

7. 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.

8. Originally Posted by Chokfull
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.

9. Originally Posted by Chokfull
(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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### volume of cylindricaql shell

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