Simpson's Rule: Estimating Dome Surface Areas

Paraboloid Section Surface of Revolution Paraboloid Section Arc Length
Dome (Paraboloid) Total Surface Area Calculation

Height = 8     Width = 20     x2 = Width/2 = 10
a = 4 × Height / Width ² = .08     Δx = (10 – 0)/4 = 2.5
1 × 0 × 2.5 × Ö (1 + 4 × .08 ² × (0 × 2.5) ² = 0
4 × 1 × 2.5 × Ö (1 + 4 × .08 ² × (1 × 2.5) ² = 10.8
2 × 2 × 2.5 × Ö (1 + 4 × .08 ² × (2 × 2.5) ² = 12.8
4 × 3 × 2.5 × Ö (1 + 4 × .08 ² × (3 × 2.5) ² = 46.9
1 × 4 × 2.5 × Ö (1 + 4 × .08 ² × (4 × 2.5) ² = 18.9
(2.5/3) × (10.8 + 12.8 + 46.9 + 18.9) = 74.5
2π × 74.5 = 468.1 ... Javascript Calculator: 467.720524

Dome Section Surface Area Calculation

x1 = 5     x2 = 7.5     Δx = (7.5 – 5)/4 = .625
1 × ( 5 + 0 × .625) × Ö (1 + 4 × .08 ² × ( 5 + 0 × .625) ² = 6.4
4 × ( 5 + 1 × .625) × Ö (1 + 4 × .08 ² × ( 5 + 1 × .625) ² = 30.3
2 × ( 5 + 2 × .625) × Ö (1 + 4 × .08 ² × ( 5 + 2 × .625) ² = 17.7
4 × ( 5 + 3 × .625) × Ö (1 + 4 × .08 ² × ( 5 + 3 × .625) ² = 40.9
1 × ( 5 + 4 × .625) × Ö (1 + 4 × .08 ² × ( 5 + 4 × .625) ² = 11.7
(.625/3) × (6.4 + 30.3 + 17.7 + 40.9 + 11.7) = 22.3
2π × 22.3 = 140.1
140.1 ÷ 6 = 23.4 ... Javascript Calculator: 23.332558

Dome Section Parabolic Arc Calculation

x1 = 5     x2 = 7.5     Δx = (7.5 – 5)/4 = .625
1 × Ö (1 + 4 × .08 ² × (5 + 0 × .625) ² = 1.28
4 × Ö (1 + 4 × .08 ² × (5 + 1 × .625) ² = 5.38
2 × Ö (1 + 4 × .08 ² × (5 + 2 × .625) ² = 2.83
4 × Ö (1 + 4 × .08 ² × (5 + 3 × .625) ² = 5.95
1 × Ö (1 + 4 × .08 ² × (5 + 4 × .625) ² = 1.56
(.625/3) × (1.28 + 5.38 + 2.83 + 5.95 + 1.56) = 3.54
... Javascript Calculator: 3.541453


Joe Bartok
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