Finding Volume: Disc Method

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This activity is for a Calculus class to help students visualize how to calculate volume by the disc method. Students will be given a typical volume problem and create 6 to 8 discs to approximate the shape. Then they make the actual shape and calculate the volume. This is the companion activity to Finding Volume: Shell Method http://www.thingiverse.com/thing:1696726 Print Settings Printer Brand: MakerBot Printer: MakerBot Replicator Rafts: Yes Supports: Yes Resolution: standard Infill: 10% How I Designed This To create the rotated object and the discs I first got the 2 dimensional graph using desmos.com. Then I did a printscreen to capture the graph [figure 1]. I used an online SVG converter and opened it in 123d Design. I used the polyline tool to enclose the region. Then I used polyline again to draw a line acting as the x axis that this shape would get rotated over [Figure 2]. Next I drew rectangles at each interval point with a small width [Figure 3]. Then I used the revolve tool to rotate each rectangle around the line I drew [Figures 4]. To get the full object I revolved the whole graph around the line [Figure 5]. I scaled both objects by .4 so that they would print smaller. figure 1 figure 2 figure 3 figure 4 figure 5 Standards NGSS CCSS Overview and Background Project: Finding Volume: Disc Method Students understand how the disc method for finding volume works and create a model of both the approximation with discs and the actual 3D figure. Students will learn how to find the volume of a region using the disc method and why the formula works. These visuals will really help students see how the rotated 3d object ends up looking. Students learn how to find the volume of a region using discs. This satisfies the AP Calculus Learning Outcome 3.4D: Apply definite integrals to problems involving area, volume. Lesson Plan and Activity The full details are in the attached worksheet. Here is a brief overview. First assign each student (or group) a region that they will be rotating around the x-axis. Then have them use desmos.com, or a similar graph plotter, to plot their region. Then they will create 6-8 discs with radius f(x). Next, they will think about the full region rotated and print that. Then they will get the formula for area of each disc and volume of the rotated region. Finally, they will calculate the volume for their region. Skills Learned volume of region Duration of Lesson The activity itself can be completed in one day but printing will take longer. Preparation This activity should be done at the point in calculus when students first learn how to find volume by discs. They probably have all of the pre-requisite knowledge, such as knowing how to evaluate integrals. You might need to remind them of the equation for area of a circle. Rubric and Assessment Here is a sample rubric to use with the worksheet. Question (points) 1 (2) 2 (10) 3 (20) 4 (5) 5 (13) Total (50) References It could be helpful to give a quick review of Riemann sums or have students watch a video on the topic. https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/riemann-sums/v/simple-riemann-approximation-using-rectangles.