A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?
Students that shaded regions can form the base of 3D objects with a cross-sectional area of a common geometric shape. Students understand that shaded regions can serve as the profile of objects rotated about the x-axis or y-axis. Students can generate integral expression for the volume of a region for any of these scenarios.
Fear Not the Y-Axis
An early introduction to integrals can make volume expressions particularly smooth. The concept of an integral is not foreign, its representation as an area is well known at this point. The new idea is that much like integrating a velocity functions yields position, integrating an area function yields volume. The integral serves to move us "up" the relationship.
Students grasped the concept fairly quickly, what we spent most of our time on was the variety of scenarios. We discuss 3D objects with square, rectangular, "leg-on" right triangle, "hypotenuse-on" right triangle, equilateral triangle, or semi-circle cross sections set perpendicular to either the x-axis or y-axis. In my first run through Calculus I ignored the y-axis to my detriment. It doesn't have to be a big deal if you don't make it a big deal.
I hand out a set of regions.
For each region, I give them three types of cross sections to consider. The y-axis regions introduce an important concept: what do I do if a part of the region is uniform in size? Seeing that a volume expression can result from two integrals will become useful later.
As students will make all kinds of notes on these pictures while working through them, I have a second set of regions ready later in the week.
This set of regions is for introducing solids of revolution. After all the mixing of shapes before, students find it relatively simple to throw this in the mix. Again I present them with three scenarios to consider for each region, and they determine the appropriate volume expression. Multiple integrals may be required.
To demonstrate understanding of volume expressions, students have be able to apply any scenario to any region. My goal here is to remove any fear of "y-axis" especially since it's not something they've been used to. Revolving a region about the y-axis should be no different than stacking a bunch of square cross sections along the x-axis.
Their (admittedly too lengthy) understanding task looked like this:
Six figures accompanied the task. None of the curves were labeled, I wanted to make a point about innate knowledge of parent functions. None of the given functions were in the appropriate form for y-axis use. I wanted to make a point about knowing how to invert something to suit your needs.
Despite the 2016 AP exam not offering a straight forward question on volume, the students exceeded my expectations on this topic. The y-axis was no big deal.
Total time spent was 5 class periods. The assessment took two 2 class periods.