Volumes of Revolution

Section 3.9 Volumes of Revolution

Subsection 3.9.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

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Subsection 3.9.2 Introduction

Volumes of revolution are often very challenging to visualize on paper. Luckily, Maple has an interactive way of visualizing the volume obtained by revolving a region about a central axis. In this activity, we will use the Volume of Revolution Tutor to find and plot the volume of a region rotated about a vertical axis or horizontal axis.

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The Volume of Revolution tutor is able to determine the volume of the solid obtained by revolution using either the disks/washers method or the method of cylindrical shells. You can learn more about how to access and use this tutor in Section 15.4.

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Exercises 3.9.3 Exercises

1.

In the first three exercises of this activity, you will be working with the region bounded by the functions \(f(x)=x^5-x^3\) and \(g(x)=\sin(x)\text{,}\) where \(x \geq 0\text{.}\)

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(a)

Assign these two functions in Maple using the assignment operator, :=.

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(b)

Plot the graphs of \(f(x)\) and \(g(x)\) on the same set of axes to view the region that you will be revolving about an axis in the next couple exercises.

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(c)

As you can see from your plot, the functions intersect at \(x_1=0\) and at another value where \(x_2>0\text{.}\) Solve for this second value, \(x_2\text{,}\) where the functions intersect and assign it to x2.

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Hint.

An example of finding intersection points is given Example 10.1.

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2.

In this exercise, you will determine the volume of the solid obtained by revolving the region between the curves \(f(x)\) and \(g(x)\) (with \(x \geq 0\)) around the horizontal line \(y=-4\text{.}\)

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(a)

Using the int() command or Int() and value(%) commands, calculate the volume of revolution using the washer method.

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(b)

Use the Volume of Revolution tutor to plot the solid and confirm your answer in part (a).

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3.

In this exercise, you will determine the volume of the solid obtained by revolving the region between the curves \(f(x)\) and \(g(x)\) (with \(x \geq 0\)) around the vertical line \(x=\pi\text{.}\)

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(a)

(Optional) If you are familiar with the method of cylindrical shells, use this method to calculate the volume of the solid using the int() command or Int() and value(%) commands

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(b)

Use the Volume of Revolution tutor to calculate the volume of the solid. Before closing the tutor, copy the text at the bottom (in the Maple Command box).

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(c)

Paste this command onto a new line and change 'output'=plot to 'output'=value to output the volume of the resulting solid.

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Hint.

You will need to include the Student[Calculus1] package by typing with(Student[Calculus1]): on a new line before the VolumeOfRevolution() command will work.

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4.

Suppose you want to find the volume of an egg that has an elliptical shape defined in the \(xy\)-plane by \(\dfrac{x^2}{2}+y^2=1\text{.}\)

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(a)

Plot the curve using the implicitplot() command.

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Hint 1.

Don’t forget to include the plots package before using implicitplot().

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Hint 2.

When plotting the ellipse, it may initially look like a circle. This is because Maple does not use the same scaling for each axis. Try clicking on the plot and using the \(1:1\) button in the top menu.

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(b)

Solve the equation of the curve for \(y\) to get the equations of the top and bottom halves of the ellipse.

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(c)

Find the \(x\)-intercepts of this ellipse.

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(d)

Use the Volume of Revolution tutor or your choice of either the int() or Int() commands to calculate the volume of the solid obtained by revolving the top half of the ellipse about the \(x\)-axis.

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