Simpson's Rule: Estimating Dome Surface Areas

Hyperbolic Arc Length Calculation
 Height = 3     Width = 13.74473     a = 3     b = 2 x1 = 0     x2 = Width/2 = 6.87387 t1 = arcsinh(0/3) = 0     t2 = arcsinh(6.87387/3) = 1.5668 Δt = (1.5668 – 0)/4 = .3917 1 × Ö ((3 cosh (0 × .3917)) ² + (2 sinh (0 × .3917)) ² = 3 4 × Ö ((3 cosh (1 × .3917)) ² + (2 sinh (1 × .3917)) ² = 13.33 2 × Ö ((3 cosh (2 × .3917)) ² + (2 sinh (2 × .3917)) ² = 8.66 4 × Ö ((3 cosh (3 × .3917)) ² + (2 sinh (3 × .3917)) ² = 24.3 1 × Ö ((3 cosh (4 × .3917)) ² + (2 sinh (4 × .3917)) ² = 8.79 (.3917/3) × (3 + 13.33 + 8.66 + 24.3 + 8.79) = 7.58 ... Javascript Calculator: 7.581137

Dome (Hyperboloid) Total Surface Area Calculation
 Height = 3     Width = 13.74473     a = 3     b = 2 x1 = 0     x2 = Width/2 = 6.87387 t1 = arcsinh(0/3) = 0     t2 = arcsinh(6.87387/3) = 1.5668 Δt = (1.5668 – 0)/4 = .3917 1 × 3 sinh (0 × .3917) × Ö ((3 cosh (0 × .3917)) ² + (2 sinh (0 × .3917)) ² = 0 4 × 3 sinh (1 × .3917) × Ö ((3 cosh (1 × .3917)) ² + (2 sinh (1 × .3917)) ² = 16.1 2 × 3 sinh (2 × .3917) × Ö ((3 cosh (2 × .3917)) ² + (2 sinh (2 × .3917)) ² = 22.5 4 × 3 sinh (3 × .3917) × Ö ((3 cosh (3 × .3917)) ² + (2 sinh (3 × .3917)) ² = 106.8 1 × 3 sinh (4 × .3917) × Ö ((3 cosh (4 × .3917)) ² + (2 sinh (4 × .3917)) ² = 60.4 (.3917/3) × (16.1 + 22.5 + 106.8 + 60.4) = 26.87 2π × 26.87 = 168.8 ... Javascript Calculator: 168.450022