# Topic 7 Applications of Integrals II

## 7.1 Volume of Revolution

In terms of definite integrals, the volume of a solid obtained by rotating a region about the \(x\)-axis can be calculated by \[\int_a^b \pi (r_1(x)^2 - r_2(x)^2) \mathrm{d} x \qquad \text{disk/washer method},\] or \[\int_a^b 2\pi r(y) h(y) \mathrm{d} y \qquad \text{shell method method},\]

where \(r_1(x)\), \(r_2(x)\) and \(r(x)\) represents the radius and \(h(x)\) represents the height of a cylindrical shell.

In practice, it’s better to recognize the shape of a cross section, find the volume of a slice of the solid and then set up the integral.

In the following, you will see some tools/commands from Maple which are very helpful to calculate the volume of a solid.

In Maple, the following command, supported by the package `Student[Calculus1]`

, can be used to get the graph, the integral and the volume of the solid obtained by rotation the region bounded by \(f(x)\), \(g(x)\), \(x=a\) and \(x=b\).

`VolumeOfRevolution(f(x), g(x), x = a..b, opts)`

To learn what options does the command `VolumeOfRevolution`

have, you may type

`?VolumeOfRevolution`

in the Math mode and hit enter. You will see the help page.

**Example 7.1**Show the solid obtained by rotating the region bounded by \(y=x^2\) and \(y=x\) about \(y\)-axis. Set up an integral for the volume. Find the volume.

*Solution. *

#Load the package

`with(Student[Calculus1])`

#Show the solid

`VolumeOfRevolution(x^2, x, x = 0 .. 1, axis = vertical, output = plot)`

#Set up an integral

`VolumeOfRevolution(x^2, x, x = 0 .. 1, axis = vertical, output = integral)`

#Find the volume

`VolumeOfRevolution(x^2, x, x = 0 .. 1, axis = vertical, output = value)`

The outputs in Maple can be seen in the following picture

*Remark. * 1. If you change the function to `VolumeOfRevolutionTutor`

, you will see an interactive popup windows which does exactly the same thing.

If the rotation axis is not an axis of the coordinate system, you need add the option

`distancefromaxis = numeric`

into the function. For example, if in the above example, the rotation is about \(y=-2\), then the Maple command should be the following`VolumeOfRevolution(x^2, x, x = 0 .. 1, axis = vertical, distancefromaxis = -2, output = integral)`

**Exercise 7.1**Find the volume of the solid obtained by rotating the region bounded by \(y=x^3\), \(x=0\), \(y=1\) about \(y\)-axis

**Exercise 7.2**Find the volume of the solid obtained by rotating the region bounded by \(y=x^3\), \(y=0\), \(x=1\) about (a) \(y=0\), and (b) \(x=2\).