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[MUSIC]. 

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Here's a little puzzle about length. 
I've got an isocoles right triangle. 

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And let's say that both of the legs of 
this right triangle have the same length 

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1. 
And we know the length of the hypotenuse 

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by the Pythagroem theorem. 
Well, that tells us that this length is 

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the square root of 2. 
But let's try to think about this 

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differently. 
What if, instead of a smooth ramp, I made 

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stairs. 
I mean, instead of this ramp picture, I 

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have a picture like this, right. 
A staircase. 

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What's the length of the staircase. 
The total length of the staircase is 2. 

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To see that, just look at the horizontal 
sections of the stairs. 

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That's the same as the entire bottom 
edge. 

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And the vertical sections of the 
staircase is the same as this whole edge. 

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So 1 plus 1 is 2. 
What happens if I make those stairs even 

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smaller? 
The length is still 2. 

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Because again, all of the vertical 
sections of the staircase add up to this 

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length. 
And all of the horizontal sections of the 

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staircase, and I'm[UNKNOWN] pushing them 
all down, add up to this length. 

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So, the total length of the staircase is 
still 2. 

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What happens if I make those stairs 
really small? 

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Well, even if the stairs are really, 
really small, the total length of the 

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staircase is still 2. 
What is going on here? 

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The length of the ramp is a square root 
of 2. 

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The length of the staircase is 2. 
No matter how small I make the stairs, 

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even if I made each stair the size of a 
proton, the staircase would still have 

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length too. 
The point is that even something as 

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seemingly obvious as length, is more 
subtle than I think we give it credit 

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for. 
What we really need is some sort of 

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definition of length. 
We'll do it in terms of an integral. 

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So just like with volume or with area, 
I'm going to take this thing that I'm 

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interested in, say the length here. 
And try to break it up into little pieces 

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that I'll then add up with an integral. 
So let's imagine that I cut this up into 

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little tiny pieces and that I just 
connect these by straight lines. 

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And then I can just add up the length of 
these little tiny straight lines and call 

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that, in the limit, after I integrate, 
the length of this curve. 

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I want to write down an integral that 
captures that intuition. 

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Well, let's just look at one little piece 
of this thing here. 

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let's write a little triangle there so i 
can call the change in x dx, and the 

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change in y dy and what I'm interested in 
is how long this little green line 

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segment is. 
Well, you might think then that little 

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green line segment has the length square 
root dx squared plus dy squared. 

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And then you integrate to get the total 
arc length, just play around with that a 

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little bit. 
Specifically lets imagine pulling a dx 

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outside of this. 
So when I get this is the intergral. 

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The square root of 1 plus, and if I 
factor out a dx here, I have to include a 

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dx in the denominator there. 
And this has the advantage at least of 

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now looking like the derivative, right, 
dy over dx, that's how I've been denoting 

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the derivative, and this form is the sort 
of thing that I can integate. 

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So here's the formula that we're going to 
use. 

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The arc length from a f of a to b f of b 
along the graph of f is the integral x 

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goes a to b of the square root of 1 plus 
the derivative squared and that's exactly 

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what I saw here dx. 
I hope that you're feeling just a little 

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bit uneasy at this point. 
I'm really treating dx and dy as if 

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they're legitimate mathematical objects, 
and you should be a little bit 

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uncomfortable with that at this point.