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:

(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 curve between x = 0 and about the x-axis.

Solution

The graph of is 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:

• 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 and into 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.