About this experiment
Stack finite-thickness layers and inspect the contribution of each one. Refine the stack to discover when layer sums are exact and how curved boundaries produce systematic approximation errors.
Learning goals
- Compute volume as section area times thickness.
- Compare a midpoint approximation with exact volume.
- Explain different error signs for cones and spheres.
How to use it
- 1Start with a cone and select a layer to inspect its midpoint area.
- 2Double the layer count and compare the signed volume error.
- 3Switch to a sphere and observe that the midpoint error has the opposite sign.
- 4Use a cuboid or cylinder and verify exact volume even with one layer.
The model
Assumptions
- Layer radii or widths are fixed at their midpoint values. They form a stepped approximation rather than exact pieces of the curved solid.
- Visual layer gaps separate the drawing only. They do not contribute to layer thickness or computed volume.
- Constant-area cuboids and cylinders are exact for every positive integer layer count.
Try this in class
- Q1Why can one cylindrical layer overestimate a sphere?
- Q2Why does doubling the number of cone layers reduce this midpoint error by a factor of four?
Frequently asked questions
Are the displayed layers exact curved slices?
No. Each is a constant-area prism or cylinder whose area equals the exact middle section. The difference from the original solid is the point of the experiment.
Does the selected layer change the total?
No. Selection highlights one contribution while the complete sum remains unchanged.