# Solid of Revolution - Shell Method

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##### Author:
Richard Enderton (273)
##### Objective:

Students should be able to calculate the volume of a solid of revolution, applying the shell method.

(more)

## Prerequisites

The student should be familiar with the concept of using the definite integral as a summation, and should also be familiar with the disk and washer methods for finding the volume of a solid of revolution.

## The Shell Method

This video develops the concept and integral formula for calculating the volume of a solid of revolution using cylindrical shells.

Source: self-created video

## Examples of applying the shell method

This video walks through several examples of applying the shell method and contains instruction on how to set up the shell method for various situations.

Source: self-created video

## Problem Set

1. Use both disk (or washer) AND shell method to find the volume of the solid of revolution if the region R is revolved about the given axis.

R is bounded by , the x-axis and x = 4.

a. Revolve R about the y-axis. (201.062)

b. Revolve R about the line x = 7. (268.083)

c. Revolve R about the line y = 10. (509.357)

d. Revolve R about the x-axis. (160.85)

2. The first quadrant region bounded by and y=4 is revolved about the y=axis.

a. Find the volume of the resulting solid. (25.133)

b. A hole is drilled in the solid. The hole is centered along the y-axis. What must the radius of the hole be to remove one-quarter of the volume of the solid? (0.732)

3. R is bounded by . Find the volume of the solid generated when R is revolved about the line x = -2. (1.517)

4. A torus (donut) has a cross section with radius 1. The center of the cross section is 4 units from the center of the torus. Use the concept of solid of revolution and the shell method to calculate the volume of this torus. (78.957)

Source: own problems

Paul’s Online Notes

Master Math Mentor

Shell Method Demo Gallery

Wolfram Alpha

### Questions and Answers

SOPHIA has reviewed the tutorial and found it academically sound.
Parmanand Jagnandan (386) - about about 1 year ago

"I really enjoyed the progression you used to justify the use of the shell method and the animations, which helped to bring the method to life."

Anthony Varela (394) - about about 1 year ago

"I really enjoyed the examples of varied axes of rotation. You provide an easy-to-follow explanation when constructing the formula around an axis of rotation that is not the x- or y-axis."

Rachel Orr-Depner (469) - about about 2 years ago

"Great examples in the second video. The first video takes a difficult concept and makes it very manageable and approachable for a student that may be struggling with the concept. "

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