Volumes of Revolution
Calculus
A classic: What will be your take on this? - Explore the mathematical principles and create your own unique investigation.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
To determine the volume of an object with circular cross-sections, we can use the idea of volumes of revolution. The topic is no longer in SLAA but is accessible and can be interesting. Fit the object with a piecewise function and model the volume on GeoGebra. Determine the curves with different methods and compare accuracy. The challenge will be to make your approach interesting and mathematically meaningful.
Guiding Questions
- Take a curve you like and rotate it around an axis. What solid do you get?
- Use the volume of revolution formula to find its volume, and check it against a known solid like a cone.
- Pick a real object — a vase, a bottle, a rugby ball. Photograph its profile, fit a function, and calculate its volume.
- Check your answer by measuring how much water the object displaces. How close are you?