Arc Length of Sine Curve

Date: 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/