Calculate volumes of solids with known cross sections using definite integrals
I can visualize solids whose cross sections are squares, rectangles, semicircles, and triangles
I can set up and evaluate definite integrals that give volumes of solids using known cross sections
Quick Lesson Plan
Food items are used in this activity to help students grasp the concept of geometric shapes sitting on a plane region. Activity questions 1 - 5 require no calculations nor numeric expressions: instead, they are designed to promote conversation among students who then develop independently the basic structure for these volume integrals. Once their conceptual understanding is confirmed in the debriefing session, the Important Ideas formalize their thinking by using definite integrals to represent volumes found by cross sections.
Topics 8.7 and 8.8 are presented after Topics 8.9 - 8.12 to ensure students are familiar with the concepts around volumes of revolution. Visualizing triangles or semicircles sitting on a plane region is much more challenging than working with disks or washers --- and the required geometric formulas often present an additional level of difficulty! Physical examples or manipulatives can be extremely helpful to students who struggle to visualize “quarter circles whose radius is the length between f(x) and g(x),” or “rectangles whose height is twice their base measured from f(x) to g(x).”
You may wish to modify this lesson by having students actually find the volume of a loaf of bread instead of using the pictures provided. A variety of baked goods with different cross sections will help clarify the differences in solids that can be created with cross sections. Help students draw out the differences between solids of revolution and solids made with cross sections on a 2-d region. Reiterate to students that the dimensions of the cross section differ depending on where it is located in the region. This reinforces why variable expressions are needed for the area of a cross section.
An excellent representation of volumes using known cross sections is found in 2008 AB1, a calculator active question. Parts (a) and (b) investigate different areas in the given region (R), but parts (c) and (d) require volume expressions. While question (c) incorporates squares as the cross sectional area, (d) introduces an additional function to represent the depth of water beneath R. Ensure adequate practice with finding points of intersection, storing those values in a calculator, and then using those points when evaluating integral expressions (see part (b)).
One of the more challenging skills required in this lesson is the ability to orient cross sections appropriately. “Each cross section perpendicular to the y-axis... “ indicates a “dy” integral with limits of integration found on the y-axis. “Squares perpendicular to the x-axis…”, conversely, represent a “dx” integral with limits of integration measured on the x-axis. Typically, the formulas for the areas of circles and equilateral triangles cause extra work for many students, but we can expect an AP Calculus student to persevere through that obstacle and then remember the formulas in the weeks to come!