# A Solid Idea

3D printing in calculus class: The shape of things to come.

One common problem in calculus is to find the volume of a solid that is created by revolving a region around an axis. The difficulty many students have in visualizing these types of shapes was the motivation behind the project that I created, with lots of help and support from the Upper School Academic Technology Coordinator, Dawn Berkeley. Groups of students were given the assignment to create several different regions using functions we have been working with in AP Calculus AB. Then, they used three-dimensional software to revolve these regions around axes. The software allows students to then rotate the shape and zoom in and zoom out to explore the shapes they created.

Imagine taking a two-dimensional semicircle and revolving it about its diameter; you would end up with a sphere. Or, if we took that same semi-circle and rotated it around a line parallel to its diameter, but a few inches away from its diameter, we’d get something resembling a donut. As the two-dimensional regions get more complicated than a simple semi-circle, so do the shapes they create when spun around an axis.

The students chose their favorite of the various shapes they created, and converted them to a file to be printed using the 3D printer. Dawn took care of scaling the models down to a reasonable size, and she and her work program students printed the models.

The final piece of the project was for students to find the actual volume of the solids that they created. After we created the models, we discussed the calculus techniques involved in finding the volume of these types of solids, and each group set about applying these techniques to their specific models.

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