All right let's get an arc length so here's my curve it's actually a special curve that
you might recognize and i'll bring that up at the end and when i'm getting arc length
i'm talking about adding up a bunch of tiny little arc length elements given by this expression
so i'm going to have to get the derivative of this curve so let's just get that done in advance
so this is really 1 minus x squared all to the one half power so i get one half
1 minus x squared to the negative 1 half power multiplied by the derivative of the interior
that's the chain rule working there and so y prime turns out to be negative x i cancel the twos over
the square root of 1 minus x squared and i'll have to square that when i put it into this expression
so my total arc length is the sum of all the contributions
so
i'm going to add all those up as x goes from negative 1 to 1 and then inside this integral i
have a 1 plus x squared over 1 minus x squared dx so a couple things about this one is that i notice
the integrand is an even function because all the appearances of x are to even powers in other words
if i replace x with negative x i get back the same integrand so that means it has mirror symmetry
across the y-axis and it could make my life a little easier by writing this this way
another thing just to illustrate some useful algebra is that i could get a
common denominator inside here by writing one as one minus x squared divided by itself
and now i can just add the numerators and when i do that i get just a 1 in the numerator
and i'm left with something that looks fairly simple
you may recognize that as an inverse sine so we end up with 2 times
the angle whose sine is x evaluated from 0 to 1. all right when i sub in the top limit i have
the angle whose sine is one well that's pi over two so two times pi over two and then when i sub
in the lower limit the angle whose sine is zero that's zero and so i end up with pi for my answer
and then if you didn't notice it at the beginning this special curve is actually a semi-circle
and so a semi-circle of radius one well a circle with radius one you would have two pi times radius
or two pi would be the circumference this arc length is half the circumference which is pi