Volumes by Slicing

Stacks of thin disks

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Volume by slicing: $V = \int A(x)\,dx$.
• Disks: $V = \int \pi\,r(x)^2\,dx$. The cone's $\frac13$ comes from integrating $x^2$.
• The napkin ring’s volume depends only on its height.

Steps

1. A sliced loaf
2. Disks
3. The cone, at last
4. Washers, shells and more
5. A satellite dish
6. The napkin ring
7. Rings from any ball