[MUSIC]. 
We defined arc length using an integral. 
The integral that we'll be using is this 
one. 
I'm going to be integrating from a to b 
the square root of 1 plus the derivative 
squared dx. 
Let's apply this to a specific problem. 
Let's compute the arc length of a little 
piece of a graph. 
So let's look at the graph of y squared 
equals x cubed and figure out the length 
of this arc from 0,0 to 1,1. 
Now I could rewrite this equation. 
Instead of writing y squared equals x 
cubed, I could write y equals x to the 3 
halves power. 
We'll start by differentiating. 
So because of this, I can write dy, dx is 
3 halves x to the 1 half power. 
We'll put that derivative into the 
formula for arc length so I'm going to be 
integrating. 
X goes from 0 to 1, the square root of 1 
plus this derivative squared. 
So, I'll write 3 halves, x to the 1 half 
squared dx. 
Now, I just want to compute that 
integral. 
But how? 
Well, I can rewrite the integrand. 
This is the integral from 0 to 1 of the 
square root of 1 plus 3 halves squared is 
9 4ths. 
And x to the one half squared, I'll write 
x, dx. 
Just have no fear, keep going. 
Well, I could use substitution. 
u equals 1 plus 9 4ths x so that du is 9 
4ths x or alternatively, dx is 4 9ths du. 
And that means this integral, it's the 
same as the integral of square root of u 
times 4 9ths du and remind you bounce of 
integration. 
Well an x is 0, u is 1 and when x is 1 u 
is 14 4ths. 
Alright, now I just have to write down an 
anti derivative here, well I could write 
this as 4 9th, u to the 3 halves over 3 
halves that's an anti-derivative for 
square root 2, evaluated at 1 and 13 4th. 
So I'm just going to plug in 13 4th plug 
in 1, take the difference. 
Okay, so this is 4 9th, I write, times 2 
3rd, instead of dividing by 3 halves. 
Times 13 4th. 
To the 3 halves power minus 4 9th times 2 
3rd times 1 to the 3 halves power and I 
got to simplify this little bit more even 
lets see here. 
4 times 2 is 8, okay that's also 4 to the 
3 halves power. 
So I can cancel all of these things. 
And then I can simplify gotta do some 
arithmetic here. 
So I've got what, 13 to the 3 halves 
power divided by 9 times 3 is 27, minus, 
this is 8 27ths just times 1. 
And, if you don't like the 3 halves here, 
I could write this as 13 times the square 
root of 13 over 27 minus 8 over 27. 
This, is the length of that arc. 
We did it, but it worked out almost too 
well. 
Alright that's the sad thing about arc 
length calculations. 
Because the arc length formula involves 
that square root of 1 plus a derivative 
squared, it's extraordinarily unlikely. 
And if you just give me some random 
function, I'm going to succeed in being 
able to evaluate that integral. 
The sorts of things we can calculate the 
arc length of are pretty special sorts of 
things.