Disc Method Volume for y = c·xⁿ

Spin the region under a curve about the x-axis and it sweeps out a solid. Slice that solid across the axis and every slice is a disc whose radius is the curve's height y, so its area is πy²; adding the discs from 0 to b is the integral of πy². For y = c·xⁿ that integral is πc²b²ⁿ⁺¹/(2n + 1). Worked example: y = √x from 0 to 4, so c = 1 and n = ½. Each disc has area πx, and the volume is π times the integral of x from 0 to 4, which is π × 8 = 25.13, or 8π cubic units. The cylinder around it, radius 2 and length 4, holds 16π, so the paraboloid fills exactly half of it, a result Archimedes proved without calculus.

The classic mistake is forgetting to square the radius. The integral of πy, rather than πy², gives π × 16/3 = 16.76 for the same solid, which is wrong by a third; the discs are circles, and a circle's area goes with the square of its radius. Two checks catch most slips: n = 0 must give a cylinder, πc²b, and n = 1 must give a cone of radius cb and height b. The same method sizes anything turned on a lathe, a nose cone, a bowl, or the paraboloid a satellite dish is cut from.