Volume of a Solid of Revolution

Figure 8.20:

Visualizing the process of finding the volume of a solid of revolution.

Overview:

This applet illustrates a technique for calculating the volume of a solid of revolution. In particular, the solid we consider is formed by revolving the curve \(y = e^{-x}\) from x = 0 to \(x = 1\) about the \(x\)-axis.

To find the volume of this solid, we first divide the region in the \(xy\)-plane into thin vertical strips (rectangles) of thickness \(Δx\). As each rectangle is rotated about the \(x\)-axis, it forms a slice that looks like a circular disk. It is easy to write the formula for the approximate radius of this disk in terms of \(x\).  (Here \(R = e^{-x}\) .) We can use this radius to state a formula for the approximate volume of this disk (slice).

\[V =\pi \left(e^{-x}\right)^2\,Δx.\]

We can approximate the total volume of the solid by adding up the volumes of a finite number of these slices (disks). As the thickness of the slices, \(Δx\), tends to zero, we get the exact volume of the solid using an integral.

This applet allows you to investigate these steps.

Instructions:

On the left there are three scrollbars. The top scrollbar allows you to vary the position of a representative rectangle within the region.

The second scrollbar allows you to rotate this rectangle about the \(x\)-axis, producing a disk. You can then use the top scrollbar again to move this disk through the region, watching to see how its radius changes as \(x\) varies.

The third scrollbar allows you to rotate the region itself about the \(x\)-axis, to form the solid of revolution.

If you click on the button labeled, Show \(n\) Slices, the interval is divided into \(n\) equal subintervals (initially \(n = 4\)), producing n rectangles in the region, and, when you rotate these using the third scrollbar, you can see the \(n\) disks formed.

After clicking on Show \(n\) Slices, a fourth scrollbar is available at the bottom of the control panel that allows you to vary the number of slices/disks, \(n\). Note that it may take a few seconds the first time you choose a new value of \(n,\) as the new 3D image is calculated. But after the first time this new image is displayed, it can be redisplayed very quickly, so eventually you can smoothly scroll through the allowed range of \(n.\)

Note: You may rotate the 3D graph at any time to get a better perspective by using the mouse to click and drag anywhere on the plot. The plot rotates about the origin in the direction you drag the mouse.

Activities & Questions

Try the following activities, in the order listed below.

1. Use the top scrollbar to vary the position of the representative rectangle through the region. Try to imagine the disk that is formed when we revolve this rectangle about the \(x\)-axis.
2. Now use the second scrollbar to actually do this. Does the disk look like you imagined it would look (in question 1)?
   Next use the first scrollbar again to move this representative slice/disk through the region.
   Try to picture the solid of revolution for which we are seeking the volume.
3. Use the third scrollbar to revolve the whole region about the \(x\)-axis forming the solid of revolution. Does the solid look as you pictured it in your mind in question 2?
4. Now move the second scrollbar back to the far left. This should remove the red disk from the picture.
   Click on the button labeled, Show \(n\) Slices. The solid is initially approximated by 4 disks of equal thickness. If the third scrollbar is still adjusted to the far right you should also see the volume of the 4 disks along with the exact total volume of the solid on the 3D plot.
   Take a few minutes to verify by hand that the total volume of these 4 disks is the quantity shown in the top line.
5. Using the fourth scrollbar at the bottom, gradually increase the number of slices/disks used in the approximation. You may want to use the 3rd scrollbar at times to look at a cutaway of these disks, but considering the approximations of the total volume shown when they are complete, can you see that these approximations get steadily closer to the actual volume?

Additional Help & Features:

Initially the region in the \(xy\)-plane, the cross section, and the solid appears opaque. To make them semi-transparent, simply click on the button labeled, Make Transparent. This allows you to see through the front surfaces to what is behind, while still maintaining a clear perception of what is in front. This mode is a little slower than drawing with solid colors. To change back to solid/opaque mode, simply hit the same button again, now labeled, Make Opaque.

The Reset button restores the applet view to its original state.

Some Useful Control Keys

	Hit Ctrl-i:		To Zoom In.
	Hit Ctrl-o:		To Zoom Out.
	Hit Home:		To restore the view to its standard viewpoint.
	Hit Ctrl-Home:		To view the solid from above the \(xy\)-plane.
	Hit the e key:		To turn the black edges in the grid on and back off.
	Hit the f key:		To turn the solid faces in the framework that forms the solid on and off.
	(When the faces are off and the edges are on, this looks like a wire mesh.)
	Hit left-arrow:		To rotate the view about the \(y\)-axis clockwise.
	Hit right-arrow:		To rotate the view about the \(y\)-axis counterclockwise.
	Hit up-arrow:		To rotate the view about the \(x\)-axis clockwise.
	Hit down-arrow:		To rotate the view about the \(x\)-axis counterclockwise.

(Note: You may have to click on the 3D plot before these control keys will work.)

This applet was created for Hughes-Hallett Calculus, published by John Wiley & Sons, by Paul Seeburger, Assistant Professor of Mathematics at Monroe Community College in Rochester, NY.