In this video I go over further into Calculus with Parametric Curves and this time derive the formula for the Arc Length, which is the length of a curve. I go over Part 1 of the arc length derivation, which looks at the specific case in which the parametric equations x = f(t) and y = g(t) can be written in the standard form y = F(x). In the derivation I use the Substitution Rule for integrals to convert the standard formula for arc length, which I have covered in my earlier videos. In the next part I show that this formula for arc length is still valid even if the curve can’t be written as y = F(x). But to do that I use polygonal approximation, which is quite amazing so stay tuned for the next video!

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Arc Length

Recall from my earlier video, that the length L of a curve C in the form y = F(x), a ≤ x ≤ b, where F' is continuous, is given by the formula:

Suppose that C can also described by the parametric equations:

This means that C is traversed once, from left to right, as t increases from α to β.

Now we can substitute the derivative dy/dx with the parametric form, as explained in my earlier video, into the arc length formula and then use Substitution Rule to obtain:

Even if C can't be expressed in the form y = F(x) (i.e. a 1-to-1 function) the above formula is still valid but we obtain it by polygonal approximations.

I will go over this in my next video so stay tuned!