# The expression of the volume of a cylindrical shell. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 6, Problem 4RCC

(a)

To determine

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

Expert Solution

The expression of the volume of the cylindrical shell is V=2πrhΔr_

### Explanation of Solution

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

Volume=(Circumference)(height)(thickness)=2πrhΔr

Here, 2πr is circumference area, h is the height, and Δr is thickness.

Therefore, the expression of the volume of a cylindrical shell is 2πrhΔr_.

(b)

To determine

Expert Solution

### 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.

V=ab(circumference)(height)(dxordy)

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

(c)

To determine

Expert Solution

### 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.

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