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
Cross Section Match-Up is a calculus version of the game Concentration as students try to collect sets of 3 related cards. Twelve cards describe a base region (defined by given functions), twelve cards describe a cross section (square, semicircle, etc.), and twelve cards contain integral expressions that will calculate the volume described on the other two cards.
This game can be played in small groups or can be used as a card sort for individual review (students working individually will likely need less time than groups). However the activity is presented, allow time up front for students to sketch the base regions as defined by the eight given functions. Some students may need a hint to create new functions by solving for x.
Cards should be placed facedown. Players will take turns turning over three cards to see if they are a match. A match consists of a region, a cross-section shape, and the correct integral expression to find the volume of the solid using that particular cross section shape. If the cards do not form a match, they are turned over again and it is the next player’s turn. This continues until all sets have been formed. The player with the most sets wins.
The recording sheet will help students keep track of their sets. This can be completed during the game or at the end. The recording sheet also defines regions Q, R, S, and T. Remind students to check each others’ work to make sure that what one player considers a match is in fact a match.
An easier (and faster) version of this game could be made by printing the regions, shapes, and integral expressions on three different colors of cardstock so students choose exactly one of each per turn.
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)).
Favorite geometry formulas make a return in this lesson! Unfortunately, students often use diameter instead of radius when finding volumes using circles, semicircles, or quarter-circles: they forget to divide the length f(x) - g(x) by 2! Reviewing properties of equilateral triangles and isosceles right triangles could be helpful as well.