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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 32
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PROFESSOR: Today we're going to continue our discussion of parametric curves. I
have to tell you about arc length. And let me remind me where we left off last time.
This is parametric curves, continued. Last time, we talked about the parametric
representation for the circle. Or one of the parametric representations for the circle.
Which was this one here. And first we noted that this does parameterize, as we say,
the circle. That satisfies the equation for the circle. And it's traced counterclockwise.
The picture looks like this. Here's the circle. And it starts out here at t = and it gets
up to here at time t = pi / 2. So now I have to talk to you about arc length. In this
parametric form. And the results should be the same as arc length around this circle
ordinarily. And we start out with this basic differential relationship. dx ^2 is dx ^2 +
dy ^2. And then I'm going to take the square root, divide by dt, so the rate of
change with respect to t of s is going to be the square root. Well, maybe I'll write it
without dividing. Just write it as ds. So this would be (dx / dt)^2 + (dy / dt)^2 dt.
So this is what you get formally from this equation. If you take its square roots and
you divide by dt squared in the inside, the square root and you multiply by dt
outside. So that those cancel. And this is the formal connection between the two.
We'll be saying just a few more words in a few minutes about how to make sense of
that rigorously. Alright so that's the set of formulas for the infinitesimal, the
differential of arc length. And so to figure it out, I have to differentiate x with respect
to t. And remember x is up here. It's defined by a cos t, so its derivative is  a sin t.
And similarly, dy / dt = a cos t.
And so I can plug this in. And I get the arc length element, which is the square root
of ) a sin t) ^2 (+ a cos t) ^2 dt. Which just becomes the square root of a ^2 dt, or
a dt. Now, I was about to divide by t. Let me do that now. We can also write the rate
of change of arc length with respect to t. And that's a, in this case. And this gets
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 Calculus

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