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### 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

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
High School Calculus

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