Tuesday, February 10, 2015

More Volumes in Calculus {Student Edition}

A couple summers ago, I made some really beautiful (I think) models to represent the kinds of figures we find the volumes of in calculus (post here).  The models worked well last year; I think it made the "formulas" make sense to the kids.  But, I thought it'd be even better if the kids actually got to create some of these in class.  I just couldn't quite figure out how I wanted to do it, without making it a project and without taking up too much class time.  And then, a year late, an idea finally came to me.

First, I bought a package of forty 5.5x8.5" foam sheets that were self-adhesive on one side ($5 at Wal-Mart).  I stacked three sheets together (so I have thirteen "boards") to produce the base of the desired solid.

Then, I graphed two functions (y=2cos(x/2) and y=e^(x/4)) on Desmos, printed them off, and used them as stencils.  So, each foam board has a graph on both sides:

I don't really know why I chose these two graphs other than the fact that I wanted one increasing and one decreasing function, both only in the first quadrant.

Next, I made and printed different kinds of cross sections for the kids to use on cardstock (see file below: squares, rectangles, semi-circles, equilateral triangles, and isosceles right triangles.

And after that, the students did the rest of the work.  They worked in groups of 2-3 to create a solid with either base f(x) or g(x) (I assigned).  Then, they calculated the volume of their solid and put their answer in this table:

Here are some examples of their finished products:

Here's how I told the kids to use the pins:

Finally, here is everything you'll need if you want to do this with your calc kids, too!

The first two pages are the two graphs I used.  The next five pages are the cross sections that I printed off on cardstock.  The graphs/cross sections are sized to fit together.  All you'll need is some foam sheets, pins, and Sharpies. :)