Before we get too deep into today’s lesson, mention the Bottle Project that students will be completing soon. I have a few projects on display in my classroom that students have noticed before, so I refer to those projects as one application of using solids of revolution to sketch the edge of a bottle and then revolve that sketched region around the centerline to create the 3D bottle. The Bottle Project also has students use cross-sectional solids to estimate the volume of their bottle, so we will wait to assign this project for a few more days until we start cross-sectional solids, but it is worth mentioning now so students know one way that they will be applying their knowledge of this topic.
Distribute the Volumes of Solids - handwritten notes and discuss each example with students. I selected these four examples to give two examples using dx and dy, and two examples using the disk and washer methods. Give students the opportunity to inspect each setup and ask questions until they fully understand why each problem was setup that way. Once students are comfortable with and understand the setups for these model questions, allow students to keep this sheet on their desk throughout today’s class as a reference whenever they might need it for today’s classwork.
Looking for Structure in Variations on the Same Problem
The In The Classroom file contains the same problem as yesterday, where students revolve the same region around 4 different lines. Project this same problem and have students flip back to yesterday’s notes to remind students of their work yesterday. Then, make 1 change to the problem such as translating the axis of revolution or one of the boundaries of the region, and have students modify their setups from yesterday to accommodate the change you made. Through this process, students should be able to determine when the limits of integration will change (when one of the boundary functions shifts a different amount than other boundary functions so as to change their intersection points) and when the disk method changes to the washer method because (basically) the axis of revolution is no longer one of the boundaries of the region. This process helps develop students’ skills in contextualizing and decontextualizing the parameters of these problems as they move back-and-forth between the geometric model and the algebraic forms of the integrals (SMP #2 reason abstractly and quantitatively) and in making use of structure (SMP #7).
With whatever time remains in this section, provide additional practice problems for students with computing volumes of solids of revolution. Make sure students are making sketches for EVERY PROBLEM, including the rectangles for the Big Radius (and the Little Radius with washers). Watch for students who confuse which function is the “Big” x or y and which function is the “Little” x or y. It can be helpful to have students think about the x-axis and the y-axis each as number lines, and ask them where the larger numbers are on each number line. The function that is further in this direction will be the “Big” function, and visa versa for the “Little” function.