Volumes in Calculus- Part 2: The Cup

This is part two of a four part series about teaching volumes in calculus. It starts here.

After reviewing the cross-section problems from the homework from last class, I pull out a can of diet Pepsi and a glass. I open the can and the whole class is watching me. I try to pretend like nothing special is going on, but they can't help but oogle at the can- I don't drink soda. I'm a health nut. And it's what my department head drinks- one per class period, as the class informs me. I tell them that I had to wrestle her for it and after hearing them go off about how bad for me it is (diet soda apparently means double calories for anything else you eat for the day) I tell them that I didn't really want to drink it anyway. All I want to know is how high up the glass it will go.

My glass is pretty much a cylinder, so I pass it around with a ruler in inches (just to really throw off all my international students). We have a discussion about measurement in inches, that's it's not 4.6 inches but 4.5 and one eighth inches. Someone finally decides to estimate the volume of the soda and the class realizes that the volume is on the can- no estimation is needed. Of course, the volume is in milliliters and then we have to figure out how to change it to cubic inches. After some texts to Google, finding converters on our cell phones, and using unit conversions, we have determined the volume to be 21.663 cubic inches and we can focus on finding the height the Pepsi will go in inches.

I pour. And our guess ends up being off by a little. Not much, but a little. So we discuss why. The Pepsi doesn't actually start at the table; there is glass in between. Still, that doesn't account for all the error. I pick up the glass and we look at it's sides. The bottom is slightly curved. They blame the error on that. If the sides were straight, they said, they could have guessed it perfectly.

So I pull out another glass, one with completely straight sides, and ask again- how high will the Pepsi go?

My students pass around the glass and all take measurements. (Note for doing this in the future- have multiple glasses of the same shape so that multiple students can be taking measurements at once.) By the time we agree on the measurements, the class is over. Their homework- figuring out how high the Pepsi will go and the next class will start off by answering this question.