University⏱ 4 min

Shell Method

Volume by cylindrical shells

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M 40.0,270.0 L 43.6,270.0 L 47.3,269.9 L 50.9,269.8 L 54.5,269.6 L 58.2,269.4 L 61.8,269.2 L 65.5,268.9 L 69.1,268.6 L 72.7,268.2 L 76.4,267.8 L 80.0,267.3 L 83.6,266.8 L 87.3,266.3 L 90.9,265.7 L 94.5,265.0 L 98.2,264.4 L 101.8,263.6 L 105.5,262.9 L 109.1,262.0 L 112.7,261.2 L 116.4,260.3 L 120.0,259.3 L 123.6,258.3 L 127.3,257.3 L 130.9,256.2 L 134.5,255.1 L 138.2,253.9 L 141.8,252.7 L 145.5,251.5 L 149.1,250.2 L 152.7,248.8 L 156.4,247.4 L 160.0,246.0 L 163.6,244.5 L 167.3,243.0 L 170.9,241.4 L 174.5,239.8 L 178.2,238.2 L 181.8,236.5 L 185.5,234.7 L 189.1,233.0 L 192.7,231.1 L 196.4,229.3 L 200.0,227.3 L 203.6,225.4 L 207.3,223.4 L 210.9,221.3 L 214.5,219.2 L 218.2,217.1 L 221.8,214.9 L 225.5,212.7 L 229.1,210.4 L 232.7,208.1 L 236.4,205.7 L 240.0,203.3 L 243.6,200.9 L 247.3,198.4 L 250.9,195.9 L 254.5,193.3 L 258.2,190.7 L 261.8,188.0 L 265.5,185.3 L 269.1,182.5 L 272.7,179.7 L 276.4,176.9 L 280.0,174.0 L 283.6,171.1 L 287.3,168.1 L 290.9,165.1 L 294.5,162.0 L 298.2,158.9 L 301.8,155.8 L 305.5,152.6 L 309.1,149.3 L 312.7,146.0 L 316.4,142.7 L 320.0,139.3 L 323.6,135.9 L 327.3,132.5 L 330.9,129.0 L 334.5,125.4 L 338.2,121.8 L 341.8,118.2 L 345.5,114.5 L 349.1,110.8 L 352.7,107.0 L 356.4,103.2 L 360.0,99.3 L 363.6,95.4 L 367.3,91.5 L 370.9,87.5 L 374.5,83.5 L 378.2,79.4 L 381.8,75.3 L 385.5,71.1 L 389.1,66.9 L 392.7,62.6 L 396.4,58.3 L 400.0,54.0r=xV=127.23
Shells: 6
V=2π∫03x⋅x2 dx=2π[x44]03=127.235V = 2\pi\int_0^3 x\cdot x^2\,dx = 2\pi\left[\frac{x^4}{4}\right]_0^3 = 127.235

Shell method: volume = sum of hollow cylindrical shells. Each shell: area = 2πr·h, thickness Δx. Useful when rotating around the y-axis.