…but what does it look like?

It will soon be time to teach about finding the volumes of solid figures using integration techniques. Here is a list of links to posts that will help your students what these figures look like and how they are generated.

Visualizing Solid Figures 1 Here are ideas for making physical models of solid figures. These make good projects for students.

A Little Calculus is an iPad app that does an excellent job in helping students visualize many of the concepts of the calculus. Volumes with regular cross section, disk method, washer method, cylindrical shells are all illustrated.

The first illustrations show square cross sections on a semicircular base. The base is in the lower part and the solid in the upper. By using the plus and minus button (lower right) you can increase or decrease the number of sections in real time and see the figures change. The upper figure may be rotated by moving your finger on the screen.

The illustration below shows a washer situation.

The following older posts show how to use Winplot to generate and explore solid figures. Unfortunately, Winplot seems to have gone out of favor. I’m not sure why; it is one of the best. I still use it and like it. You may download Winplot here for free (PC only).

Visualizing Solid Figures 2 This post demonstrates how to use Winplot to generate solids with regular cross sections and solids of rotation.

Visualizing Solid Figures 3 The washer method is illustrated using Winplot. These post all relate to finding volumes by washers: Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures 4 Using Winplot to see the method cylindrical shells. Note that this method is not tested on either the AB or BC Calculus exams, so you do not have to teach it. Many teachers present this topic after the exams are given. As a footnote you may also find Why You Never Need Cylindrical Shells interesting. (However, this is not the reason it is not tested on the AP Calculus exams.)

Visualizing Solid Figures 5 An exercise demonstrating how “half” can mean different things and shows that how the figures are generated makes a difference.

Visualizing Solid Figures 2

You have probably caught on by now that Winplot is my favorite computer graphing program. In addition to being great at drawing quick graphs, it is able to produce and rotate 3D images of, among other things, solids of rotation, and solids with regular cross-sections. In this post I will discuss how to do solids of regular cross-section and solids of rotation. In my next posts I’ll show you how to see the disks, washers, and shells.

Winplot is a free program. Click here for Winplot for PC and here for Winplot for Macs. (May 11, 2017 Note: Winplot is no longer available from its original home. The link for PCs above connect to another site where the program can be downloaded. For Macs use the PC link, but use the Winplot for Macs link for instructions and another program you will need.You can also Google Winplot and find other sites that have the program as well as many, many instructional videos.)

Solids with regular cross-sections

Consider the region bound by the graphs of f\left( x \right)=\sqrt{x-1} and g\left( x \right)=\tfrac{1}{2}\left( x-1 \right) from x = 1 to x = 5.

Begin by opening a Winplot 2D graphing window, graphing the curves, and adjusting the window to a good scale. Use the box where the equations are entered (Equa > 1.Explicit) check “lock interval,” and enter the “low x” and “high x” values (1 and 5 respectively) to stop the graphs where they intersect. Click “ok” to see the graphs.

Solids 2 A

On the navigation bar, click on “Two” and then “Sections.” You should see a window like this:

Solids 2 B

The top two drop-down boxes at the top allow you to choose which curves to work with, and since we have only two they should already be selected. Then click on the cross-section shape you want – square, equilateral triangle, or semicircle. The box below that allows other shapes where the height may be set (the height(x) may be  a number or a function of x). Set the “low x” and “high x” to the left and right sides of the region. Then click “see solid” and you will see the solid in a new window.

Click on the new 3D window and then type Ctrl+A to show the axes. Rotate the image by using the 4 arrow keys, and zoom in and out with the Page Up and Page Down keys.

Solids 2 CNow let’s get fancy. Close the 3D window and return to the cross-section box shown above. Change the “high x” to 5@B (you may use any almost letter except x, y, or z). Then click “see solid.” Next, in the 3D Window click Anim > Individual > B. This will give you a slider. Slide the slider from 1 to 5 and you will see the solid grow and see the square cross-sections. (The video uses the “autocyc” button – use S to slow the animation, F to speed it up and Q to quit.)

Square x-sections

Use File > Save As… to save the image. It will save with the extension .wp3 and you will lose the original 2D graphs. The animation buttons will still work when you open it again.

Solids of Revolution.

Solids of rotation are done in a similar way. We will revolve the same curves around the horizontal line y = –1.  Enter the curves as above and click on One > Revolve Surface.  Curves are revolved one at a time, so choose the first curve from the drop-down box. Choose the axis the figure is to be rotated around by entering the values for a, b, and c in ax + by = c, or clicking on one of the axis buttons.  For the “arc start” and “arc stop” values use the left and right ends of the region. The “angle start” and “angle stop” values are the default, 0 and 2pi (entered as “2pi”). Again we have made this last value 2pi@A so that we can animate the graph.

Solids 2 D

Click “see surface” to see the revolved surface.  As before, use the 4 arrow keys and the Page Up and Page Down keys to adjust the image, and Ctrl+A to show the axes.

Surfaces are revolved one at a time so return to the “surface of revolution” window and use the drop-down box to choose the next curve. Leave all the other values the same. Clicking “see surface” will graph the second curve with the first and show the solid figure. Note that the surfaces are graphed in the same color as the original 2D graphs.

Solid rotation

Use the slider or autorev or autocyc buttons to watch the curves revolve. (Remember to type “F” to speed up the motion. “S” to slow it down, and “Q” to quit.)

The next posts will show how to see the disks, washers, and shells, and animate them along with the surfaces.

Volume of Solids with Regular Cross-sections

Applications of Integration 2 – Solids with Regular Cross-sections

In Vino Veritas. And not only truth, but the start of an important application of the calculus. After Johannes Kepler’s (1571–1630) first wife died he decided to marry again. Naturally, he bought wine for the festivities. There was some disagreement with the wine merchant concerning how he calculated the volume of the barrel of wine. This led Kepler, a mathematician, to study the problem of finding the volume of barrels. The barrels, then as now, had circular cross sections of different diameters. His idea was to consider the wine as a stack of thin cylinders (with height of as we would say today). He then calculated the volume of each cylinder and added them together to find the volume of the barrel. There is a nice animation of the idea and more information on Kepler here.

The basic problem is this: you have a solid object whose volume you need to compute. If you can slice the object perpendicular to the axis in such a way that you get regular cross sections (i.e. similar figures) whose area you can compute, then by multiplying the area times the thickness and adding the results, you can have the volume. This kind of addition is done nicely by integration, since the volume of the slices form a Riemann sum.

You need to set up a coordinate system so that you can slice the figure into thin slices (whose width is ) and express the cross-section’s dimensions as a function of x and then the area as A(x). The result always looks like this where A(x) is the area of the cross-section:

\displaystyle \underset{n\to 0}{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{A\left( {{x}_{i}} \right)\Delta {{x}_{i}}}=\int_{a}^{b}{A\left( x \right)dx}

Think of this as the integral of the cross-section area times the thickness

Since you are probably not allowed to have wine in your calculus class, you may want to illustrate the idea some way other than using a wine barrel. Think of a loaf of sliced bread, or a deck of cards. Find some object that the students can measure and calculate the volume, such as a tree trunk, a statue, or a Coke-Cola bottle.

Another interesting exercise is to develop (prove) the formulas for volumes that students have known and used without proof since grade school: a cone, a pyramid, a sphere, the frustum of a cone.