Difference between revisions of "Length of a curve using numerical methods"
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Instead of evaluating this integral with a table, a numerical method called Simpson's rule can be used: | 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<sub>n</sub> = (<span>Δ</span>x/3)(f(x<sub>0</sub>) + 4f(x<sub>1</sub>) + 2f(x<sub>2</sub>) + 4f(x<sub>3</sub>) + 2f(x<sub>4</sub>) + ...+ 2f(x<sub>n-2</sub>) + 4f(x<sub>n-1</sub>) + f(x<sub> | + | the length of the curve after n iterations = L<sub>n</sub> = (<span>Δ</span>x/3)(f(x<sub>0</sub>) + 4f(x<sub>1</sub>) + 2f(x<sub>2</sub>) + 4f(x<sub>3</sub>) + 2f(x<sub>4</sub>) + ...+ 2f(x<sub>n-2</sub>) + 4f(x<sub>n-1</sub>) + f(x<sub>n</sub>)) |
where <span>Δ</span>x = (b-a)/n with a and b being the boundaries and n the number of iterations of the calculation. | where <span>Δ</span>x = (b-a)/n with a and b being the boundaries and n the number of iterations of the calculation. | ||
Revision as of 19:33, 1 April 2021
Simpson's Rule is one of many methods 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 = Ln = (Δx/3)(f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ...+ 2f(xn-2) + 4f(xn-1) + f(xn))
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 heating liquid 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
which is very close to the number obtained using integral tables (77 degrees). The value will get closer as the number of iterations is increased.
