Arc Length of Sine CurveDate: 7/18/96 at 17:13:14 From: Anonymous Subject: Arc Length of Sine Curve For years I have been trying to find the arc length of one cycle of a standard sine wave. That is the arc length from zero to two pi of sin x. I have had some people tell me it was an elliptical integral; others say it is not, but in any case it is a bear. Any suggestions? Date: 7/19/96 at 12:17:34 From: Doctor Paul Subject: Re: Arc Length of Sine Curve The formula for arc length is the integral from a to b of sqrt(f'(x)^2 + 1) The derivative of sin(x) is cos(x). Square it, add one, take sqrt and get: sqrt(diff(sin(x),x)^2+1); 2 1/2 (cos(x) + 1) Now integrate this from zero to 2*Pi: (This has no elementary antiderivitive so I'm using Maple to help me) Int(sqrt(diff(sin(x),x)^2+1),x=0..(2*Pi)); 2 Pi / | 2 1/2 | (cos(x) + 1) dx | / 0 then tell Maple to evaluate this integral and return a number: evalf(Int(sqrt(diff(sin(x),x)^2+1),x=0..(2*Pi)),10); 7.640395578 And there's your answer. -Doctor Paul, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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