Volume by Cylindrical Shells

A slecture by Hamdaan Khalid

Partly based on the MA 166 (calculus 2) Fall 2017 lecture material.

1. Introduction

In this tutorial, I shall explain calculation of volume by cylindrical shell. The reason why I’ve chosen this particular topic is because, unlike the disk and washer method, most students have a difficulty calculating volume by cylindrical shells. I myself had to put in hours of work, but finally understood the concept and ended up making it my preferred choice for questions involving volume in calculus 2. The explanation given below will help you visualize, and apply the cylindrical shell method in your classes.

2. Concept (Formula and Visualisation)

The formula for calculating volume through cylindrical shells is given below:

1) Rotation about $ x $-axis $ \displaystyle \int_a^b 2\cdot \pi\cdot R\cdot H \,dx $

2) Rotation about $ y $-axis $ \displaystyle \int_a^b 2\cdot \pi\cdot R \cdot H\, dy $

In both cases ‘$ R $’ refers to the "radius" if the graph were to be revolved around the given axis, “$ H $” is given as the “height” of the shape if it were to be revolved around the given axis, and “$ a $” and “$ b $” are the bounds on the axis we are asked to rotate the graph upon.

To go about using the given formula in an actual problem, all you need to be able to do is identify the radius and the height. The approach to identifying the radius and the height in these problems is to visualize the graph move. If we are revolving around the “$ x $-axis", the height will be extending on the $ y $-axis, and the radius will be on the $ x $-axis. If we are revolving around the “$ y $-axis", the height will be extending on the $ x $-axis, and the radius will be on the $ y $-axis.

In all the questions, you will be given one variable as the function of the other, identify if the given function is the radius or the height, and put it in place of “$ H $” or “$ R $” in the format it is written in. Use the other variable in place for “$ R $” or “$ H $”, whichever one is unused, this variable will be put in as “$ x $” or “$ y $”. For the bounds, look at the axis the graph is being rotated on and identify from where to where it extends on the $ x $ or $ y $ axis.

For questions involving a shape enclosed by two functions being rotated, identify the radius or height as either, the function in the right minus the one on the left or the function above minus the function below.

In case we are asked to revolve the graph around $ x=2 $ or $ y=-1 $ or any such given situation, we have to be able to visualize how does the radius or height change with respect to the axis of revolution.

In the video tutorial I will be going all of what I have discussed so far and will also be solving a few examples.

Watch the video tutorial and then attempt to solve the recommended problems from the video.

Link to Video on Youtube

3. References


Questions and comments

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