Chapter 6, Problem 4RCC

(a)

To determine

To Find: The expression of the volume of a cylindrical shell.

Answer to Problem 4RCC

The expression of the volume of the cylindrical shell is $\underset{\_}{V=2\pi rh\Delta r}$

Explanation of Solution

The volume of the cylindrical shell is the product of the circumference area, height, and thickness of the cylindrical shell.

$\begin{array}{c}\text{Volume}=\left(\text{Circumference}\right)\left(\text{height}\right)\left(\text{thickness}\right)\\ =2\pi rh\Delta r\end{array}$

Here, $2\pi r$ is circumference area, h is the height, and $\Delta r$ is thickness.

Therefore, the expression of the volume of a cylindrical shell is $\underset{\_}{2\pi rh\Delta r}$.

(b)

To determine

To define: how to find the volume of a solid of revolution using of cylindrical shells.

Explanation of Solution

The region revolved by the rectangles is forms cylindrical shells rather than disks or washers.

After the revolution find the circumference and height of the solid revolution in terms of x and y for a typical shell.

Consider the shell revolved within the limit a and b.

If the rectangle revolved about x-axis then the radius is dy.

If the rectangle revolved about y-axis then the radius is dx.

$V={\displaystyle {\int }_a^b\left(\text{circumference}\right)\left(\text{height}\right)\left(dx\text{or}dy\right)}$

Therefore, the expression of the volume of a solid of revolution using of cylindrical shells is explained.

(c)

To determine

To define: shell method is usable method instead of slicing.

Explanation of Solution

The slicing method to find the solid of revolution produces disks and washers sometimes whose radii or difficult to find explicitly. But in the cylindrical shell method forms an easier integral method to find the volume.

Therefore, the shell method is usable method instead of slicing.