[MUSIC]. 
Here's a little puzzle about length. 
I've got an isocoles right triangle. 
And let's say that both of the legs of 
this right triangle have the same length 
1. 
And we know the length of the hypotenuse 
by the Pythagroem theorem. 
Well, that tells us that this length is 
the square root of 2. 
But let's try to think about this 
differently. 
What if, instead of a smooth ramp, I made 
stairs. 
I mean, instead of this ramp picture, I 
have a picture like this, right. 
A staircase. 
What's the length of the staircase. 
The total length of the staircase is 2. 
To see that, just look at the horizontal 
sections of the stairs. 
That's the same as the entire bottom 
edge. 
And the vertical sections of the 
staircase is the same as this whole edge. 
So 1 plus 1 is 2. 
What happens if I make those stairs even 
smaller? 
The length is still 2. 
Because again, all of the vertical 
sections of the staircase add up to this 
length. 
And all of the horizontal sections of the 
staircase, and I'm[UNKNOWN] pushing them 
all down, add up to this length. 
So, the total length of the staircase is 
still 2. 
What happens if I make those stairs 
really small? 
Well, even if the stairs are really, 
really small, the total length of the 
staircase is still 2. 
What is going on here? 
The length of the ramp is a square root 
of 2. 
The length of the staircase is 2. 
No matter how small I make the stairs, 
even if I made each stair the size of a 
proton, the staircase would still have 
length too. 
The point is that even something as 
seemingly obvious as length, is more 
subtle than I think we give it credit 
for. 
What we really need is some sort of 
definition of length. 
We'll do it in terms of an integral. 
So just like with volume or with area, 
I'm going to take this thing that I'm 
interested in, say the length here. 
And try to break it up into little pieces 
that I'll then add up with an integral. 
So let's imagine that I cut this up into 
little tiny pieces and that I just 
connect these by straight lines. 
And then I can just add up the length of 
these little tiny straight lines and call 
that, in the limit, after I integrate, 
the length of this curve. 
I want to write down an integral that 
captures that intuition. 
Well, let's just look at one little piece 
of this thing here. 
let's write a little triangle there so i 
can call the change in x dx, and the 
change in y dy and what I'm interested in 
is how long this little green line 
segment is. 
Well, you might think then that little 
green line segment has the length square 
root dx squared plus dy squared. 
And then you integrate to get the total 
arc length, just play around with that a 
little bit. 
Specifically lets imagine pulling a dx 
outside of this. 
So when I get this is the intergral. 
The square root of 1 plus, and if I 
factor out a dx here, I have to include a 
dx in the denominator there. 
And this has the advantage at least of 
now looking like the derivative, right, 
dy over dx, that's how I've been denoting 
the derivative, and this form is the sort 
of thing that I can integate. 
So here's the formula that we're going to 
use. 
The arc length from a f of a to b f of b 
along the graph of f is the integral x 
goes a to b of the square root of 1 plus 
the derivative squared and that's exactly 
what I saw here dx. 
I hope that you're feeling just a little 
bit uneasy at this point. 
I'm really treating dx and dy as if 
they're legitimate mathematical objects, 
and you should be a little bit 
uncomfortable with that at this point.