What do cucumbers and calculus have in common? I use them both to introduce the topic of calculating volumes of rotated solids by disks.
I divided my calculus class into pairs and gave each pair of students a plate, a knife, and a cucumber. We reviewed the formula for the volume of a cylinder, and we then discussed how we might figure out the volume of their cucumbers. Since they aren’t perfect cylinders, using the formula alone won’t give a very good approximation. However, if we slice them into disks, each disk is very cylindrical, and we can add up their volumes to get the total volume. To check your results, you can drop the cucumber slices into a beaker of clean water and measure the displacement.
Calculus (integration, to be specific) takes the same process and adds up an infinite number of disks, which provides us with the exact volume of a rotated solid. All you need is a function that provides an accurate cross section, and an axis of rotation.
I think my students’ favorite part is when we break out the ranch dressing and eat the cucumber disks!