Length of a curve using numerical methods - Revision history

Milllo at 02:45, 2 April 2021

2021-04-02T02:45:25Z

2021-04-02T01:46:00Z

2021-04-02T01:33:41Z

2021-04-02T01:32:14Z

2021-04-02T01:31:48Z

2021-04-02T01:31:11Z

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2021-04-02T01:03:07Z

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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 integrals 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>&#916;</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>))

← Older revision		Revision as of 01:46, 2 April 2021
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	= 73 degrees		= 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.	+	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, which can easily be done using a computer.

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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<sub>n</sub> = (<span>&#916;</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>0</sub>))
+the length of the curve after n iterations = L<sub>n</sub> = (<span>&#916;</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>&#916;</span>x = (b-a)/n with a and b being the boundaries and n the number of iterations of the calculation.

← Older revision		Revision as of 01:32, 2 April 2021
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−	Simpson's Rule is used to calculate integrals numerically.	+	Simpson's Rule is one of many methods used to calculate integrals numerically.

	From the [[Length of a curve\|Length of curve page]], the length can be calculated using the integral: <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx		From the [[Length of a curve\|Length of curve page]], the length can be calculated using the integral: <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx

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-The length of curve equation used in the example was <font size = "+2"><span>&#8747;</span></font>sqr(1+(30e<sup>−0.3t</sup>)<sup>2</sup>)dt
+The length of curve equation used in the heating liquid example was <font size = "+2"><span>&#8747;</span></font>sqr(1+(30e<sup>−0.3t</sup>)<sup>2</sup>)dt
 using boundary conditions of 0 and 5 minutes, and using 10 iterations, the problem becomes:

← Older revision		Revision as of 01:31, 2 April 2021
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	= 73 degrees		= 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.

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 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))
+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 [https://www.symbolab.com/solver/function-arithmetic-composition-calculator/f%5Cleft(x%5Cright)%3D%5Csqrt%7B%5Cleft(1%2B%5Cleft(30e%5E%7B-0.3x%7D%5Cright)%5E%7B2%7D%5Cright)%7D%2C%20f%5Cleft(5%5Cright) 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)

← Older revision		Revision as of 01:27, 2 April 2021
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	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))		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))

−	= (1/6)(1 +	+	= (1/6)(1 + 425.84 + 222.247 + 419.155 + 216.495 + 414.206 + 212.238 + 410.546 + 29.091 + 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

← Older revision		Revision as of 01:04, 2 April 2021
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	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))		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))
		+
		+	= (1/6)(1 +

← Older revision		Revision as of 01:03, 2 April 2021
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	the subintervals are {0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5}		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)(f(0) + 4f(1/2) + 2f(1) + 4f(3/2) + 2f(2) + 4f(5/2) + 2f(3) + 4f(7/2) + 2f(4) + 4f(9/2) + f(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))