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[MUSIC]. 

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We defined arc length using an integral. 
The integral that we'll be using is this 

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one. 
I'm going to be integrating from a to b 

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the square root of 1 plus the derivative 
squared dx. 

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Let's apply this to a specific problem. 
Let's compute the arc length of a little 

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piece of a graph. 
So let's look at the graph of y squared 

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equals x cubed and figure out the length 
of this arc from 0,0 to 1,1. 

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Now I could rewrite this equation. 
Instead of writing y squared equals x 

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cubed, I could write y equals x to the 3 
halves power. 

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We'll start by differentiating. 
So because of this, I can write dy, dx is 

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3 halves x to the 1 half power. 
We'll put that derivative into the 

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formula for arc length so I'm going to be 
integrating. 

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X goes from 0 to 1, the square root of 1 
plus this derivative squared. 

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So, I'll write 3 halves, x to the 1 half 
squared dx. 

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Now, I just want to compute that 
integral. 

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But how? 
Well, I can rewrite the integrand. 

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This is the integral from 0 to 1 of the 
square root of 1 plus 3 halves squared is 

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9 4ths. 
And x to the one half squared, I'll write 

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x, dx. 
Just have no fear, keep going. 

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Well, I could use substitution. 
u equals 1 plus 9 4ths x so that du is 9 

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4ths x or alternatively, dx is 4 9ths du. 
And that means this integral, it's the 

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same as the integral of square root of u 
times 4 9ths du and remind you bounce of 

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integration. 
Well an x is 0, u is 1 and when x is 1 u 

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is 14 4ths. 
Alright, now I just have to write down an 

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anti derivative here, well I could write 
this as 4 9th, u to the 3 halves over 3 

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halves that's an anti-derivative for 
square root 2, evaluated at 1 and 13 4th. 

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So I'm just going to plug in 13 4th plug 
in 1, take the difference. 

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Okay, so this is 4 9th, I write, times 2 
3rd, instead of dividing by 3 halves. 

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Times 13 4th. 
To the 3 halves power minus 4 9th times 2 

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3rd times 1 to the 3 halves power and I 
got to simplify this little bit more even 

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lets see here. 
4 times 2 is 8, okay that's also 4 to the 

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3 halves power. 
So I can cancel all of these things. 

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And then I can simplify gotta do some 
arithmetic here. 

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So I've got what, 13 to the 3 halves 
power divided by 9 times 3 is 27, minus, 

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this is 8 27ths just times 1. 
And, if you don't like the 3 halves here, 

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I could write this as 13 times the square 
root of 13 over 27 minus 8 over 27. 

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This, is the length of that arc. 
We did it, but it worked out almost too 

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well. 
Alright that's the sad thing about arc 

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length calculations. 
Because the arc length formula involves 

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that square root of 1 plus a derivative 
squared, it's extraordinarily unlikely. 

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And if you just give me some random 
function, I'm going to succeed in being 

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able to evaluate that integral. 
The sorts of things we can calculate the 

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arc length of are pretty special sorts of 
things.