Are you a frustrated calculus teacher? Do you constantly find yourself on the verge of losing your tenure because your students just cannot grasp the concept of using integrals to find the volume of a shape made by using cross sections perpendicular to the x-axis between two curves? Do you just really love calculus and want a cool model of this concept?
Well look no further than the Calculus Integral Cross Section Volume Problem Model! This mouthful of a visual aid will solve all of your calculus class issues and fulfill your desire for a really big math paperweight. It fits in a 25x25x25cm cube (based on the size of the axes).
This model was printed with a 1.0mm nozzle at 0.5mm layer height in order to maximize strength and speed, so tolerances may be different for traditional 0.4mm nozzles and 0.2mm layer heights.
Overview and Background
After experiencing the chaos that was trying to visualize this concept in an AB Calculus class, I decided we could design a model for this problem that would help my teacher and other calculus teachers throughout the world. After a meeting with my teacher, we came up with the rough sketches seen below and decided to have sets of square and semicircle cross sections to show how the problem can change depending on how the integrals relate to the area formula of the cross sections.
We initially only had an x and y axis with the two functions snapping into the axes and each other where intersections occurred. This allowed us to print the parts in different colors separately. We added a z axis by drilling a 6.5mm diameter hole at the origin of the axes part (the z axis already has the hole in the design) and screwing a bolt with about the same diameter (6.4-6.45mm) through the parts to connect them.
One problem we had to troubleshoot was the issue of tight fits with the cross sections going into the two functions. With such a small indent, the pieces needed to be almost the size of the holes themselves in order to fit without falling out. One may be able to get a better fit by using a more precise 0.4mm nozzle and 0.2mm layer heights.
A rough sketch done during a brainstorming session with the customer
A demonstration of the lesson plan and assembly of the model