Simpson's Rule: Estimating Dome Surface Areas

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