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Jotaf

Simple Calculus problem: integrating over a line segment in 3D

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I probably should know this, but somehow I can't be sure of the answer to this problem. Surely it will be considered trivial by some of you in this forum :) I have a function f(x,y,z) in 3D space - the domain is R^3, and it returns a value in R. I need to find out how to integrate this function over a line segment, defined by 2 points. If you can just help me with this, I'll deal with the rest myself (actually integrating the function I have), so don't worry about that! I just want some pointers on how someone would go about doing this integration with any function. My hypothetical solution might be terribly wrong, so I won't post it for now, just in case someone gives me a quick reply. Thanks in advance!

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Guest Anonymous Poster
I don't really know much about this stuff but I think you can represent your line as a parametric function and then express the value of the line in the 3d function as a composite function, and then integrate that. I think that would work but I really haven't done this stuff so...

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f(x,y,z) is your function
a and b are the points.

the line segment between a and b may be parametrized as s(t) = (1-t)a + tb, t in [0,1].

So, to integrate you must do:

I(f(s(t))*norm(ds/dt)*dt) = I(f(s(t))*norm(a-b)dt), from 0 to 1.

I is the integrate signal.

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read up on line integrals in a calculus book. are you solving this numarically?

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Your problem is sampling the curve in space. Google for "Integration by quadrature" and "Monte Carlo integration".

-cb

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