Length of a curve using numerical methods

Simpson's Rule is used to calculate integrals numerically.

From the Length of curve page, the length can be calculated using the integral: ∫sqr(1+f '(x)²)dx

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Instead of evaluating this integral with a table, a numerical method called Simpson's rule can be used:

the length of the curve after n iterations = L_n = (Δx/3)(f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ...+ 2f(x_n-2) + 4f(x_n-1) + f(x₀))

where Δx = (b-a)/n with a and b being the boundaries and n the number of iterations of the calculation.

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The length of curve equation used in the example was ∫sqr(1+(30e^−0.3t)²)dt

using boundary conditions of 0 and 5 minutes, and using 10 iterations, the problem becomes:

g(t) = sqr(1+(30e^−0.3t)²

Δt = (5-0)/10 = 1/2

the subintervals are {0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5}

and the answer is thus = (1/6)(g(0) + 4g(1/2) + 2g(1) + 4g(3/2) + 2g(2) + 4g(5/2) + 2g(3) + 4g(7/2) + 2g(4) + 4g(9/2) + g(5)) a great resource for doing these calculations is here

= (1/6)(1 + 4*25.84 + 2*22.247 + 4*19.155 + 2*16.495 + 4*14.206 + 2*12.238 + 4*10.546 + 2*9.091 + 4*7.841 + 6.768)

= (1/6)(1 + 103.36 + 44.494 + 76.62 + 32.99 + 56.824 + 24.476 + 42.184 + 18.182 + 31.364 + 6.768)

= (1/6)(438.262)

= 73 degrees