Originally posted by wormwoodyeah-but you take 2 different routes, one is 100miles longer, but you use the same amount of fuel.
you're probably getting confused with the 'line' part? but the thing is, it's a method of doing something, not some kind of a line that you could visualize.
if you drove a car over some mountains, I think you maybe could use a line integral to calculate the amount of fuel needed to drive along a specific route.
or you go for a little round trip and lo! you have used no fuel!
gah...
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Originally posted by geniusIn a "normal" intergral, you are summing a functions value over an axis from that coordinate system (the x-axis for instance). In this case you end up with the area under the function.
i encountered them (line integrals) last year in vector calculus, and this year i've got them in complex analysis. but does anyone actually know what they are? it's been bugging me for a while. like, how would you visualise one?
my lecturer informed me that they are "abstract tools used to assist with real integrals", or something along those lines, but th erly much. but it did a bit...
anyone got any better ways of explaining them?
please?
The function you are intergrating is some way of relating x to y (or whatever) in some specific way for your problem. So if "y" is force and "x" is distance, your intergral gives you work done.
The line intergral is defined exactly the same, but now you are summing the value of a function along some path in space. So in three dimensions, the function you are summing takes a specific value at each point in 3D space (it's a field). The path you are actually intergrating over is just a curve through this field.
So if your field is a map of air pollution and you want to get from A to B with the least damage to your health, then the line intergral would provide you with a way of summing this amount of pollution along different possible paths.
Hope that actually makes some sense!
Nick
Originally posted by nickhawkerThe thing that bothers me about these kind of integrals is this:
In a "normal" intergral, you are summing a functions value over an axis from that coordinate system (the x-axis for instance). In this case you end up with the area under the function.
The function you are intergrating is some way of relating x to y (or whatever) in some specific way for your problem. So if "y" is force and "x" is distance, your interg ...[text shortened]... pollution along different possible paths.
Hope that actually makes some sense!
Nick
If you have some kind of randomness as to the curve, doesn't that mean you have to chop the integration into pieces that you can use a recognizable function to integrate a smaller piece of? If I just draw a random squiggle on a board and have to integrate it, either normal or line, it would seem you would have to do it in sections. Is this correct?