Finding the Distance of a Polynomial Function: Help Needed

[watty08]

Hello
I have a function, and need to find the distance of that polynomial function.
Not sure how to do this any help.
Couldnt figure how to put the formulas up so just put the link up i found on here.
https://www.physicsforums.com/showthread.php?t=100423
Is this the right formula and how do i use it.

 

[HallsofIvy]

You mean the length of the graph, not "distance of a polynomial function"?.
How, exactly are you given the function? Is this in two or three dimensions? Are you given the function as a single equation or as parametric functions.

Yes, the general idea of that link is correct. If, in two dimensions, you are given y= f(x), then the length of the graph from x= a to x= b is given by
$$\int_a^b \sqrt{1+ (f'(x))^2} dx$$

If you are given parametric equations, x= x(t), y= y(t), the length of the graph from t₀ to t<sub>1 is
$$\int_{t_0}^{t_1}\sqrt{(x'(t))^2+ (y'(t))^2}dt$$

In three dimensions, a single equation will not define a curve but if you have parametric equations, x= x(t), y= y(t), z= z(t), the length of the graph from t0 to t1 is
$$\int_{t_0}^{t_1} \sqrt{(x'(t))^2+ (y'(t))^2+ (z'(t))^2}dt$$</sub>

 

[Feldoh]

Use Ivy's arc length function:

If we assume that the function in continuous on [a,b] then the distance between any two points is $$\sqrt{x^2+y^2}$$ and if we shrink this to an infinitesimal length then $$\sqrt{dx^2+dy^2}$$. Next we want to integrate these distances over a to b then sum them up so:

$$\int_a^b \sqrt{dx^2 + dy^2}$$

Factor out a dx^2 essentially...

$$\int_a^b \sqrt{1+ \frac{dy^2}{dx^2}} dx$$

And notice the second term in the integral is just f'^2, so we obtain the desired result:

$$\int_a^b \sqrt{1+ (f'(x))^2} dx$$

 

[watty08]

I think its a degree four, it might be a three, i have five points.
Starting fron the origin, (0,0) Point O, Point A (2,-2), Point B (6, 2.15) point c (10,-3) Point E (16,7).
Im not really good at this stuff.
Put it in my calculator and get this
a= 6.555e-03
b=-0.1810119
c=1.44681547
d=-3.2220238
e=5.902e-11
and
r^2=1
Below it, it says
y=ax^4+bx^3+cx^2+dx+e
so i figure i sub a,b,c,d,e into that, and that how i get my function

y=6.555e-03x^4 - 0.1810119x^3 + 1.44681547x^2 - 3.2220238x + 5.902e-11