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gordonj005

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Hello, this will be my first post on the physics forum, so i wanted to make it decent :P

I've been trying recently to derive for myself a path integral formulation (not quantum mechanical or anything feynman like but for finding the length of a curve on a given interval). Heres my attempt at deriving it for a simple curve, but its a very time consuming tedius process to actually calculate. I know real path integrals involve Reinman sums, but after trying to learn them on my own, I decided to give it a shot on my own, here goes:

f(x) = x^2, from x1 = 3 to x2 = 4 using a small interval dx = 0.01

distance between two points on a function is defined as:

d = √(x2 - x1)^2 + (f(x2) - f(x1))^2

so breaking the curve down into infentesibly small portions, (or in this case our small interval dx), and adding the distances together, we should get a reasonably good approximation for the distance on the curve... right?

so in this case, between x1 = 3 and x2 = 4, we have several peices of the curve, ex, the distance between 3 - 3.01, 3.01 - 3.02, 3.02 - 3.03...3.99 - 4.0. I've worked it out so that the number of these peices is defined by (x2 - x1)*(1/dx). so in this case we have:

1*100 , so 100 individual peices we have to sum up. now this is where I've run into some problems. I've defined the sum as:

Ʃ(i = 1 to 100) = √((3 + i*dx) - (3 + (i-1)*dx))^2 + (f(3 + i*dx) - f(3 + (i-1)*dx))^2

which gives a reasonable approximation. now my problem is, i want to evaluate this sum as dx tends to zero, so we would have an exact answer. so we could write this as:

lim(dx -> 0) Ʃ(i = 1 to ∞) = √((3 + i*dx) - (3 + (i-1)*dx))^2 + (f(3 + i*dx) - f(3 + (i-1)*dx))^2

but i am stuck here, i feel like i should take an integral, but i don't know how and that probably isn't correct (or even doable). which leads me to my question, what now?

thank you very much

EDIT: using 0.01 i got ~6.94, and continuing to decrease it it seems to converge at ~7.074

I've been trying recently to derive for myself a path integral formulation (not quantum mechanical or anything feynman like but for finding the length of a curve on a given interval). Heres my attempt at deriving it for a simple curve, but its a very time consuming tedius process to actually calculate. I know real path integrals involve Reinman sums, but after trying to learn them on my own, I decided to give it a shot on my own, here goes:

f(x) = x^2, from x1 = 3 to x2 = 4 using a small interval dx = 0.01

distance between two points on a function is defined as:

d = √(x2 - x1)^2 + (f(x2) - f(x1))^2

so breaking the curve down into infentesibly small portions, (or in this case our small interval dx), and adding the distances together, we should get a reasonably good approximation for the distance on the curve... right?

so in this case, between x1 = 3 and x2 = 4, we have several peices of the curve, ex, the distance between 3 - 3.01, 3.01 - 3.02, 3.02 - 3.03...3.99 - 4.0. I've worked it out so that the number of these peices is defined by (x2 - x1)*(1/dx). so in this case we have:

1*100 , so 100 individual peices we have to sum up. now this is where I've run into some problems. I've defined the sum as:

Ʃ(i = 1 to 100) = √((3 + i*dx) - (3 + (i-1)*dx))^2 + (f(3 + i*dx) - f(3 + (i-1)*dx))^2

which gives a reasonable approximation. now my problem is, i want to evaluate this sum as dx tends to zero, so we would have an exact answer. so we could write this as:

lim(dx -> 0) Ʃ(i = 1 to ∞) = √((3 + i*dx) - (3 + (i-1)*dx))^2 + (f(3 + i*dx) - f(3 + (i-1)*dx))^2

but i am stuck here, i feel like i should take an integral, but i don't know how and that probably isn't correct (or even doable). which leads me to my question, what now?

thank you very much

EDIT: using 0.01 i got ~6.94, and continuing to decrease it it seems to converge at ~7.074

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