Visualizing Solids of Revolution

A project two of my friends and I did for our Multivariable class, showing different visualizations, examples, and applications of finding the volume of a solid of revolution using disks, washers, and shells - a concept we learned last year. 

Apples and Solids of Revolution

Here's an idea of how to visualize calculating the volume of solids of revolution with disks. You'll need a sliceable object of your choice - I chose an apple - as well as a ruler and a knife. The goal will be to cut up the apple into increasingly more slices and show that the more disks you have, the more accurate your volume estimate is. 

Before slicing up the apple, I found its volume via displacement to be 121 cubic centimeters. It was (about) 5 cm high, so I marked off 1 cm slices. 

Then I cut the apple in half, measured the radius, and calculated the volume as if the apple were a single cylinder/disk with a height of 5 cm. Obviously this is going to be a gross overestimation of the volume, as the apple is clearly not a cylinder, but we can improve our estimate by slicing again. 

With two cuts, you now have two different radii. Unfortunately, the disks have different heights, though this could be fixed if you instead opted for a slicing method in which you take each previous slice and cut it in half (resulting in exponentially more slices). 

Conceptual drawing of an apple as approximated by cylindrical disks

In the end, the approximate volumes I got were as follows: 

1 cut (2 disks) - 171.05972 cm³ 

2 cuts (3 disks) - 165.95463 cm³

3 cuts (4 disks) - 157.76031 cm³

4 cuts (5 disks) - 126.77023 cm³

The approximations nicely approach the true volume of 121 cm³. Unfortunately, it becomes quite difficult to cut thinner cylinders accurately (unless you happen to be a master fruit chopper ninja). You could instead opt to continue your apple slicing on the computer - simply take a picture and measure the slice radii virtually. This particular image shows slices every quarter centimeter. 

The idea you're illustrating here is that as you use more and more disks, the approximation approaches the true volume. If you were to cut the apple into infinitely many (infinitesimally thin) slices and add them up, you'd get the exact volume, which is what integrals allow you to do. 

Alternatively, you could take this apple slicing even further and plot some points along the edge of the apple, solve for the equation of the polynomial going through all of those points, and then do the actual integral. I however, am going to go eat these apple slices. 

Optimization Problem via Geometry

So I was browsing my old calculus book (don't judge...yes, I browse my calculus book), and I saw this optimization problem -

Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when θ1=θ2.

I'm pretty sure I even did this question, but now looking back on it the answer seems obvious. 

Because if you simply take either triangle PQR or RST and reflect it across line QT (the ground), now when you're looking for the shortest length of rope between points P and S, you know from geometry that this is just a straight line. Then you don't even need to do any calculus because θ1 and θ2 are vertical angles so they're congruent. 

What bothered me a lot about most of my elementary and middle school math was that you would be taught how to do a certain kind of problem and then expected to follow that method exactly - there was never room for exploration or creativity or anything and an answer was somehow "wrong" if we didn't do it "the book's way." Personally, I think both the calculus and the geometry paths are equally valid and satisfying, just two different ways of telling the same story.