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?

>>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.