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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 31
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PROFESSOR: OK. Now, today we get to move on from integral formulas and methods
of integration back to some geometry. And this is more or less going to lead into the
kinds of tools you'll be using in multivariable calculus. The first thing that we're going
to do today is discuss arc length. Like all of the cumulative sums that we've worked
on, this one has a storyline and the picture associated to it, which involves dividing
things up. If you have a roadway, if you like, and you have mileage markers along
the road, like this, all the way up to, say, sn here, then the length along the road is
described by this parameter, s. Which is arc length. And if we look at a graph of this
sort of thing, if this is the last point b, and this is the first point a, then you can think
in terms of having points above x1, x2, x3, etc. The same as we did with Riemann
sums.
And then the way that we're going to approximate this is by taking the straight lines
between each of these points. As things get smaller and smaller, the straight line is
going to be fairly close to the curve. And that's the main idea. So let me just depict
one little chunk of this. Which is like this. One straight line, and here's the curved
surface there. And the distance along the curved surface is what I'm calling delta s,
the change in the length between, so this would be s2 - s1 if I depicted that one.
So this would be delta s is, say s. si - si - 1, some increment there. And then I can
figure out what the length of the orange segment is. Because the horizontal distance
is delta x. And the vertical distance is delta y. And so the formula is that the
hypotenuse is delta x ^2 + delta y ^2. Square root. And delta s is approximately
that. So what we're saying is that delta s ^2 is approximately this. So this is the
hypotenuse. Squared. And it's very close to the length of the curve.
And the whole idea of calculus is in the infinitesimal, this is exactly correct. So that's