Volume

Volume of Solid of Revolution

(You need Java Runtime Environment (JRE) to run GraphFunc applet in this website)

Demonstrate some examples of finding the volumes of solids of revolution about the x-axis. Then use the GraphFunc utility online to confirm the results. (See demo)

Let y = f(x) represent a smooth curve on the closed interval [a, b], as shown in Figure 1. The volume of solid of revolution obtained from revolving the region bounded by f(x) from x = a and x = b about the x-axis is

Figure 1

Example Find the volume of the solid formed by revolving the region bounded by the curve y = x from x = 0 to x = 3 about the x-axis. Use the GraphFunc utility online to confirm the result.

Solution

The graph of y = x is shown in Figure 2. The Figure 3 shows the solid is formed by revolving y = x about the x-axis.

We have

()

Figure 2

Figure 3

Use the GraphFunc utility online to verify the above result as shown in the following steps:

Begin by going to http://graph.seriesmathstudy.com, and wait for the applet to be loaded into your computer.

(Notice that you need JRE installed in your computer first before using this website.)

Enter the expression x at command line and click on the Graph It! button.

Select the View 3D item from the drop-down list.

Select the x-axis from the drop-down list.

Enter the limit values x = 0 and x = 3 into the text fields: .

Click on the Volume button to compute the volume of the solid as shown in Figure 4.

Figure 4

We see that the volumes that GraphFunc computed and we derived are the same.

Example Find the volume of the solid formed by revolving the region bounded by the curvebetween x = 0 andabout the x-axis.

Solution

The graph ofis shown in Figure 5. The Figure 6 shows the solid is formed by revolving f(x) about the x-axis.

()

Figure 5

Figure 6

Use the software GraphFunc online to verify the above result as shown in the following steps:

Begin by going to http://graph.seriesmathstudy.com/ , and wait for the applet is loaded into your computer.

Enter the expression sin(x) at command line and click on the Graph It! button.

Select View 3D item from the drop-down list.

Select x-axis item from the drop-down list.

Enter the limit values x = 0 andinto the textfields: .

Click on the Volume button to compute the volume of the solid as shown in Figure 7.

Figure 7

We see that the volumes that GraphFunc computed and we derived are the same.

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