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I am having trouble remembering some of the material required for my current calculus course so I am reviewing some of the previous material that I have forgotten.

I am having trouble following the definition ofThe Arc Length Functionas presented in James Stewart's "Calculus: Fourth Ed." page 579.

I already follow how to derive the formula for arc length. But I am having problems with the concept of the arc length function. I am given the following:

Now I am not following the purpose of this section on arc length function. I can already figure out (5) with ds being equal to (3). So what is the purpose of this section? And I am also not following what the text is doing at the beginning by replacing the variable of integration with t and then differentiating and re-replacing t with x again. What the hey? I would really like to know what they are doing in this step. We will find it useful to have a function that measures the arcl ength of a curve from a paraticular starting point to any other point on the curve. This, if a smooth curve c has the equation y = f(x), a =< x =< b, let s(x) be the distance along C from the intial point Psub0(a,f(a)) to the point Q(x,f(x)). The s is a function, called the arc length function, and, by the formula

L = integral of (1+[f'(x)]^2)^1/2 dx

then

(1) s(x) = integral of (1+[f'(t)]^2)^1/2 dt

with the limits of [a,b].

(We have replaced the variable of integration by t so that x does not have two meanings.) We can use Part 1 of the Fundamental Theorem of Calculus to differentiate (1) (since the integrand is continuous):

(2) ds/dx = (1+[f'(x)]^2)^1/2 = (1+ (dy/dx)^2)^1/2.

Equation (2) shows that the rate of change of x with respect to x is always at least 1 and is equal to 1 when f'(x), the slope of the curve, is 0. The differential of arc length is

(3) ds = (1+ (dy/dx)^2)^1/2 dx

and this equation is somtime written in the symmetric form

(4) (ds)^2 = (dx)^2 + (dy)^2.

The geometric interpretation of equation (4) can be used as a mnemonic device for remembering (1). If we write

(5) L = integral ds,

then from equation (4) we can solve to get (3).

Any help is appreciated. Thankyou.

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# Definition of Arc Length Function

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