Solids of Revolution

This simulation approximates the volume of a solid of revolution with thin discs, each of volume pi times f of x squared times dx. Students choose a curve such as square root of x, sine of x, a cone or a sphere, choose left, midpoint or right sampling, and set the number of discs from 1 to 100 or let Auto-add run. Readouts show the number of discs, the sum of discs, the exact volume and the error. Teachers can use it to introduce Riemann sums and the disc method in calculus.

• Start with 8 Discs and read the Error readout.
• Raise Discs to 100 and watch the sum approach the Exact volume.
• Compare Left, Midpoint and Right under Sample at for the same number of discs.
• Choose the sphere curve and check the exact volume is 32 pi over 3.
• Turn off Show exact solid to see only the discs.

1. 1Explain how thin discs approximate the volume of a solid of revolution.
2. 2Compare left, midpoint and right Riemann sampling.
3. 3Observe that the approximation converges to the definite integral as discs increase.
4. 4Apply the formula V = pi times the integral of f squared.

Questions teachers ask