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I am very pleased to announce the publication of a new ebook, “Introduction to Calculus”, by Wolfram Media:

ABSTRACT (original book): This ebook provides a comprehensive introduction to fundamental concepts in calculus and their applications, covering all of Calculus 1 and some of Calculus 2. The book starts with functions and limits, followed by differential calculus, and then moves on to integral calculus and a brief discussion of differential equations. It ends with the complete solution for a sample exam based on the AP Calculus AB course. Videos for the topics covered in the ebook, along with practice problems, exercises and quizzes, are available in the interactive Wolfram U course Introduction to Calculus. CITATION (original book): John Clark and Devendra Kapadia (2024), Introduction to Calculus, Wolfram Media, Inc. ISBN: 978-1-57955-080-6 (digital online). https://www.wolfram-media.com/products/introduction-to-calculus/

The book covers all the topics in the popular Wolfram U course on Calculus and provides a comprehensive introduction to fundamental concepts in calculus and their applications.
The goal is to help students who prefer to learn Calculus using a convential textbook format.
You can download the ebook for free at the Wolfram Media website:
https://www.wolfram-media.com/products/introduction-to-calculus/
This is the first book in the “Wolfram ETextbook Series” and we hope to publish around 15 such textbooks during the next two years, including books on Linear Algebra, Finite Mathematics, Complex Analysis, etc.
Many thanks to all the teams at Wolfram and the members of Wolfram Community for the strong support and encouragement given by them for our online courses, study groups and other activities.

Below is a sample lesson from the book

Volumes by Cylindrical Shells

Overview

You previously found the volume of solids using the disk and washer method. The cross sections were disks and washers, respectively.

For example, the volume of the solid made by rotating the function

f[x]=

around the

axis from 1 to 4 is given:

Integrate[π(x^2)^2,{x,1,4}]

1023π

And the plot is this:

RevolutionPlot3D[x^2,{x,0,5},RevolutionAxis"X",BoxRatios{1,1,1}]

With the washer method, the main idea is to take the volume of one disk and subtract the volume of another disk.

However, consider the following function:

Plot[4x^3-x^4,{x,0,4}]

If you rotate it around the

axis, it is not clear what the inner and outer radii are.

The goal of this lesson is to find a way to find the volume in such circumstances with the cylindrical shell method.

Intuition

To find the volume, you will still divide the solid into pieces, but the pieces will now be shells instead of cross sections. Each shell will look like a cylinder, which is why the method is called the cylindrical shell method.

Here is a plot to illustrate:

RevolutionPlot3D[{1,t},{t,0,1}]

The volume of the given shell is

h-π

h=2πh

(

)

(

)

for

≥

. If you let



, then their difference gets small (call it

r

) and their average becomes

Therefore, letting

, the volume becomes

2πhrr

Let

the function be the height, and make it a function of

. If you add up multiple shells, the volume becomes

∫2πrf[r]r

Example

Now look back to the introduction.

You wanted to find the volume of the solid made by rotating the following function from 0 to 4 around the

axis:

f[x_]:=4x^3-x^4

Here is a plot of the solid with

RevolutionPlot3D

RevolutionPlot3D[f[x],{x,0,4},RevolutionAxis"Z",BoxRatios{1,1,1}]

With the cylindrical shell method, the volume is

∫

2πr(4

)r

Integrate[2πr(4r^3-r^4),{r,0,4}]

4096π

Washer versus Cylindrical Shell

Find the volume of the solid made by rotating the region enclosed by

and

about the

axis.

Here is the region in the

plane:

Plot[{x^2,x^3},{x,0,1},Filling{1{2}},PlotLegends"Expressions"]

	2 x
	3 x

With the washer method, the volume is

∫

π









y

Integrate[π(y^(2/3)-y),{y,0,1}]

With the cylindrical shell method, the volume is

∫

2πx(

)x

Integrate[2πx(x^2-x^3),{x,0,1}]

The volume is the same with both methods, so it is up to you to determine which to use.

Positive Axis

Suppose you rotated the following function in the range

[0,4]

about the line

x=6

f[x_]:=x^2-2x+2

Here it is with

RevolutionPlot3D

RevolutionPlot3D

f[x+6]	-6<=x<=-2
Null	True

,{x,-6,6},RevolutionAxis->{0,0,1}

Look at the 2D plot of the function:

Negative Axis

Look at the 2D plot of the function:

Washer over Cylindrical 1

Sometimes the washer/disk method is preferable to the cylindrical method. Consider the following function:

Find the volume using the cylindrical shell method.

The height of each shell will now go left to right instead of top to bottom as shown in this 2D plot:

The shell radius goes from top to bottom instead of left to right.

Washer over Cylindrical 2

Both the cylindrical shell method and washer/disk method have their advantages and disadvantages. It is up to you to determine which is better suited to each problem.

Summary

The cylindrical shell method gives another way to find the volume of solids.

The method works by dividing the solid into many cylindrical shells with infinitesimal width and height a given function.

Sometimes the cylindrical shell method can find volumes that the disk/washer method would make too cumbersome/impossible.

Other times, the disk/washer method is more intuitive and easier to use than the cylindrical shell method.

The next lesson will use the integration techniques to calculate the average value of a function.

Exercises

Exercise 1—Trigonometric Function

Exercise 2—Positive Axis

Find the volume of the solid using the cylindrical shell method.

Exercise 4—Volume of a Sphere

Exercise 5—Pontoon

Here is a graph of the pontoon:

Find the volume of the pontoon using the cylindrical shell method.

CITE THIS NOTEBOOK

Introduction to Calculus: fundamental concepts and applications with videos and course
by John Clark and Devendra Kapadia
Wolfram Community, STAFF PICKS, April 29, 2024
https://community.wolfram.com/groups/-/m/t/3166670