Circumference by integral?

So,I was playing with integrals of sin(x),and I noticed that the integral of sin(x)+1 form -pi/2 to 3pi/2 is 2pi.So I assumed that this result has something to do with the circumference of the unit circle.Then I noticed that you can remove the sinus form the function,because the integrals cancel out (-cos(-pi/2)-(-cos(3pi/2))=0),so i was left with the integral of 1.And becAuse the radius of the unit circle is 1,I put in other number,and got the correct circumferences.Next,I tried to compute the length of the archimedean spiral doing a single revolution,putting x instead of a constant in the integrated function,and got pi^2.I can't find any info on what this length really is,so I'm here now and asking this-is it the correct length,and is this formula (integral of r from -pi/2 to 3pi/2) of any value,for example,in calculating the circumferences of elipses?

>>1
First things first, definitely do it from 0 to 2pi, absolutely no point doing it between the limits you did and it just complicates the matter.

So what you're trying to work out is arc length.
I'm not going to go into derivations here, but reference http://en.wikipedia.org/wiki/Arc_length

Basically for a function defined in polars co-ordinates, which makes your two examples simpler, the arc length between two angles w is the integral of sqrt(r^2 + (dr/dw)) dw

luckily in your two cases this works out easily.

For 1. r=1 for all w and so you get the integeral of 1 from 0 to 2pi

For your second case r=w so (dr/dw)=1.


therefore you want to calculate the integral of sqrt(w^2+1) dw, which you should probably be easily able to do by a substitution of w=cosh(y) or something similar [maybe tan(y) would be better, you decide]

Or just change variables to polars.

>>2
OP here.
 I'm sorry-I just graduated high school,So i have no idea how to do this integration by substitution-I read some course notes for fun,but that's it.

P.S:Group theory looks interesting.

if you look at a circle you can define arc length as s=r(theta)

so lets say we take a very small piece of arc length

ds=r*d(theta)

so now we have a differential equation in effect and can do the integration.

int(ds,s)[0]-[Circumfrance]=r*int(d(theta),theta)[0][2pi]

Circumfrance = r*2Pi

>>4
if you're interested in group theory, you can start studying it without having taken other math stuff

it's a very fundamental branch of math, and starts from the ground up, so you dont need to know any multivariable calc, linear algebra, differential equations, or any of the usual stuff you learn before group theory.

if you pick up an intro to group theory book (or pdf) and go from beginning through sylow's theorems, that should be about equivalent to an intro course in group theory.

for free textbooks, gigapedia.org is your friend, just be sure to use the gigapedia search engine and not google's/yahoo (those dont return any results).