Chapter 3.9, Problem 31E

(a)

To determine

To find: The formula for the approximate volume of the thin cylindrical shell with height h, inner radius r and thickness $\Delta r$.

Answer to Problem 31E

The formula for approximate value of the cylindrical shell is $\Delta V\approx 2\pi rh\Delta r$.

Explanation of Solution

Calculation:

The volume of the cylindrical shell with r is the radius and h is height is $V=\pi r^{2}h$.

Here, the value h is constant and the value r is changeable variable.

The differential is $dV=f^{\prime }\left(r\right)dr$.

The derivative of the function $f\left(r\right)=\pi r^{2}h$ is computed as follows.

$\begin{array}{c}{f}^{\prime }\left(r\right)=\frac{d}{dr}\left(\pi r^{2}h\right)\\ =\pi h\frac{d}{dr}\left(r^2\right)\\ =2\pi rh\end{array}$

Substitute $f^{\prime }\left(r\right)=2\pi rh$ in $dV=f^{\prime }\left(r\right)dr$.

$dV=2\pi rhdr$

The thickness of the cylindrical shell is $\Delta r$. That is, $dr=\Delta r$.

$\Delta V\approx 2\pi rh\Delta r$

Therefore, the formula for approximate value of the cylindrical shell is $\Delta V\approx 2\pi rh\Delta r$.

(b)

To determine

To find: The error involved in using the formula from part (a).

Answer to Problem 31E

The error is $\pi h{\left(\Delta r\right)}^2$.

Explanation of Solution

Calculation:

The volume of the cylindrical shell with r is the radius and h is height is $V_1=\pi r^{2}h$.

The volume of the cylindrical shell with radius is $r+\Delta r$ and the height h is $V_2=\pi {\left(r+\Delta r\right)}^{2}h$^.

The volume of the thin cylindrical shell with height h, inner radius r and thickness $\Delta r$ is computed as,

$\begin{array}{c}\delta V=V_2-V_1\\ =\pi {\left(r+\Delta r\right)}^{2}h-\pi r^{2}h\\ =\pi h\left(r^2+2r\Delta r+{\left(\Delta r\right)}^2-r^2\right)\\ =2\pi rh\left(\Delta r\right)+\pi h\left({\left(\Delta r\right)}^2\right)\end{array}$

The error $\delta V-\Delta V$ is computed as follows.

$\begin{array}{c}\delta V-\Delta V=2\pi rh\left(\Delta r\right)+\pi h\left({\left(\Delta r\right)}^2\right)-2\pi rh\Delta r\left(\Delta V\approx 2\pi rh\Delta r\right)\\ =\pi h{\left(\Delta r\right)}^2\end{array}$

Therefore, the error is $\pi h{\left(\Delta r\right)}^2$.