- #1
wubie
Hello,
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 of The Arc Length Function as 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.
Any help is appreciated. Thankyou.
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 of The Arc Length Function as 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:
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).
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.
Any help is appreciated. Thankyou.