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 xaxis. 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 xaxis 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 xaxis. 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 xaxis.
We have
() 
Figure 2
Figure 3


(Notice that you need JRE installed in your computer first before using this website.)
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 xaxis.
Solution
The graph ofis shown in Figure 5. The Figure 6 shows the solid is formed by revolving f(x) about the xaxis.
() 
Figure 5
Figure 6 

Figure 7
We see that the volumes that GraphFunc computed and we derived are the same.

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