[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.