Finding Length of Line Traced by f(x)

In summary, there are formulas for finding the length of a line traced out by a function, such as the formula for a polynomial of degree less than or equal to 2 and the theorem for a function with a continuous derivative. However, for polynomials of degree greater than or equal to 3, the length cannot be computed exactly and numerical methods must be used. The length can also be defined as the limiting process of approximating small line segments, and can be evaluated using Simpson's rule.
  • #1
Alkatran
Science Advisor
Homework Helper
959
0
Are there any formulas for finding the length of the line traced out by some function f()?

IE: If f(x) = cx + b where c and b are constants
The length from 0 to x is:
l(x) = sqr(x^2 + (cx)^2)

But I don't know what to do for any polynomial above a line.
I can make a summation for it, but don't know how to simplify.

l(x) = lim[t->infinity](sum[n = 0 to t](sqr( 1/x^2 + (f(x*n/t)-f(x*(n+1)/t)))^2)))

Basically the sum of arbitrarily small linear approximations.


I figure the length of sin() and cos() are related to pi somehow...
 
Mathematics news on Phys.org
  • #2
As written in my calculus textbook:

Theorem: Let f be afunction s.t. f ' is continuous on [a,b]. The length L of the curve joining the points R(a,f(a)) and S(b,f(b)) is given by

[tex]L=\int_a^b\sqrt{1+(f '(x))^2}dx[/tex]
 
  • #3
For a polynomial y=y(x) of degree larger or equal to three,the integral cannot be computed exactly.U'd be dealing with so-called "LEGENDRE ELLIPTIC INTEGRALS".
Only numerical methids would work.

Daniel.
 
  • #4
Alkatran said:
Basically the sum of arbitrarily small linear approximations.
This is exactly how the length is defined if you take the limiting process.
I'll use slightly different notation then you did.

A small line segment of width [itex]\Delta x[/itex] can be approximated by:
[tex]\sqrt{(\Delta x)^2+(\Delta y)}[/tex]

by cutting up the interval [a,b] into n subintervals of width [itex]\Delta x[/itex], you can approximate the length by:

[tex]L \approx \sum_{i=1}^n\sqrt{(\Delta x_i)^2+(\Delta y_i)}=\sum_{i=1}^n\sqrt{1+(\frac{\Delta y_i}{\Delta x_i})^2}\Delta x_i[/tex]

The approximation gets better of n gets larger.
The length of the curve is defined by:

[tex]L=\lim_{n \to \infty}\sum_{i=1}^n\sqrt{1+(\frac{\Delta y_i}{\Delta x_i})^2}\Delta x_i=\int_a^b \sqrt{1+y'(x)^2}dx[/tex]

The square root often makes it very difficult or impossible to evaluate explicitly. We'll have to resort to approximating the length. With Simpson's rule for example.
 
Last edited:

1. How do you find the length of a line traced by f(x)?

To find the length of a line traced by f(x), you will need to use the arc length formula: L = ∫√(1 + (f'(x))^2) dx. This formula takes into account the changing slope of the function at each point and calculates the length of the curve.

2. Can the length of a line traced by f(x) be negative?

No, the length of a line cannot be negative. This formula calculates the distance traveled along the curve, which is always a positive value. If the function has negative values, the arc length formula will still calculate a positive length.

3. What is the difference between arc length and the length of a line traced by f(x)?

Arc length is the distance along a curve, while the length of a line traced by f(x) is the distance traveled along the curve in a specific interval. The arc length formula takes into account the changing slope of the function, while the length of a line traced by f(x) only considers the start and end points of the interval.

4. Can the length of a line traced by f(x) be calculated for any type of function?

Yes, the arc length formula can be used to calculate the length of a line traced by any continuous function. However, for some complex functions, the integral may be difficult or impossible to solve analytically, in which case numerical methods can be used to approximate the length.

5. How can finding the length of a line traced by f(x) be useful in real-world applications?

The length of a line traced by f(x) can be used in various fields such as engineering, physics, and biology to describe the path of motion of an object or the shape of a curve. It can also be used to calculate speed, acceleration, and other important parameters in these fields.

Similar threads

  • General Math
Replies
1
Views
716
  • General Math
Replies
2
Views
686
  • General Math
Replies
4
Views
771
Replies
2
Views
1K
  • General Math
Replies
12
Views
1K
  • General Math
Replies
1
Views
687
Replies
1
Views
720
Replies
1
Views
737
  • General Math
Replies
9
Views
1K
  • General Math
Replies
3
Views
760
Back
Top