Volume by “Washers”

Today I will show you how to visualize not just the solid figures but the disks and washers used in computing the volume using Winplot. The next post will show how to draw shells.

Winplot is a free program. Click here for Winplot and here for Winplot for Macs.

For the example we’ll use the situations from the 2006 AP calculus exams question AB1 / BC 1. The students were given the region between the graphs of y = ln(x)  and y = x -2. In the first part they were asked to find the area of the region. To do that they first had to determine, using their calculator, where the curves intersect. The x-coordinates of the intersections  are x = 0.15859 and x = 3.14619.

In part (b) they were asked to find the volume of the solid formed when the region was rotated around the horizontal line y = -3 . The volume is found by using the disk/washer method. Here is how to show the washers using Winplot. This gets a little complicated so I will mark each step with a bullet

• Starting in the 2D window, graph the two functions as shown in the previous post.. When entering the equations click the “lock interval” box and enter 0.159 for “low x” and 3.146 for “high x.”
• Next we will enter a Riemann sum rectangle which we will be able to move, and, once rotated, will appear as the washer. Go to Equa > Segment > (x,y) and in the box enter the endpoints of the vertical segment between the two graphs in terms of B: x1 = B, y1 = ln(B), x2 = B, and y2 = B – 2. Click “ok.”
• Go to the Anim button, choose “B” (Anim > Individual  > B).
• Enter the left value 0.15859 and click “set L”, and enter 3.14619 and click “set R.” (Remember how to do this, as we will do it again.)
• You may now move the “Riemann rectangle” (which, of course, is very thin, approaching 0) across the region.

Next we will produce the 3D images.

• As we did in the last post click on One > Revolve surface… Enter the values shown below. (The “arc start” and “arc stop” value are the x-values of the intersection points. Attach an “@S” to the “angle stop” as shown.)

• Click “see surface.”
• In the 3D window that appears click Anim > S and you will be able to revolve the curve. Make the “set L” value -2pi by typing the value in the box and clicking “set L,” leave “set R” at 2pi. Adjust the value to 0 by typing 0 and “enter.”
• Adjust the viewing widow with the 4 arrow keys and the Page Up and Page down keys. Add the axes with Ctrl+A.
• Return to the “surface of revolution” window and choose the second function from the drop-down box at the top. not change anything else. Click “see surface” and the second curve will be added to the graph.

Next we graph the “washer:”

• In the surface of revolution box, select the segment in the drop-down box at the top change the “angle stop” to 2pi@R. Click “ok.”
• Then in the 3D Inventory window for this file select the segment and click “edit.”
• Change the “low t” value to 0 and the “high t” value to 1. Change the “u hi” to 2pi@R. Click “ok.” The window should look like the one below.

Finally, in the 3D window:

• To show the line y = –3, in the 3D window go to Equa > 2. Parametric and enter the values shown in the box below and click “ok.” A short segment at y = -3 will appear in the 3D window.
• In the 3D window go to Anim > Individual and open a slider for “B” and for “R.”
• For the “B” slider make “set L” = 0.15859 and the “set R” to 3.14619 (the intersection values).
• For the “R” slide make “set L” to 0 and “set R” to 2pi.
• Adjust the R and S sliders to 0 and the B slider to its minimum value.
• Save everything just to be safe. The extension will be “.wp3.” Later you can open this file from the 3D window, but it will no longer be in touch with the 2D window even if you save that.

That should do it.

Move first the B slider, then the R slider, then the B slider again and finally the S slider to explore the situation.

In the video at the top you will see this example with these things happening in order.

• The Riemann rectangle moving in the plane using the B slider
• The Riemann rectangle rotated into a washer using the R slider.
• The washer moving through the curves using the B slider again.
• The two curves rotated part way using the S slider
• The washer moving through the solid using the B slider.
• The solid rotated with the 4 arrow keys.

The next post will show how to do a similar animation for the cylindrical shell method.