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March 30th, 2019, 03:07 PM   #1
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Check my understanding (line integrals)

2019-03-30 (1).jpg

Ok so if I'm understanding correctly, line integrals integrate over a curve (C) unlike over a region on the x or y axis like a regular integral would.

So you have a curve represented by the vector function r(t) = C = x(t) i + y(t) j

let's take a region along that curve that can be defined by the parameter (t) interval: [a,b]


now let's divide this region on curve C into sub-intervals (see attachment picture)

so now if you want to integrate over the region we have along the curve C, using our sub-intervals, we can evaluate a function (this is where I'm a little iffy about being right or not) of two variables that exists along our region on curve C at a point within each sub-interval along the region on curve C, then multiply that by the arc length of the sub-interval in which our point in located in, and add up all the values you get for different points in each sub-interval, which forms a Riemann sum that is similar to the Riemann sum formed by approx. a regular integral.


So if everything I stated above is correct, then I have a question about something. In my textbook, it says the point you choose to use when you evaluate the function that exists on curve C must be within each sub-interval along C. However, If I'm getting this correctly, this really isn't something that applies when actually computing the line integral (like in a HW or test problem) because you'll be assuming that the function f(x,y) you are given in a typical line integral problem will indeed exist on the curve C you are given. Is that correct?

Last edited by skipjack; March 30th, 2019 at 06:06 PM.
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April 8th, 2019, 12:19 PM   #2
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The type of line integral you are describing is sometimes referred to as a line integral with respect to arc length and might be written $\int_C f(x,y)~ds$ A typical application might be where $f(x,y)=1$ in which case the integral would give the length of the curve. Another might be where $f(x,y) = \delta(x,y)$, where $\delta$ gives the mass/length density and the integral gives the mass of the wire. There are also line integrals with respect to $x$ or $y$, which look like $\int_C f(x,y)dx + g(x,y)dy$ which are useful in applications like calculating calculating work done while moving along a curve in a force field.
A nice introduction to the subject is located here:
Calculus III - Line Integrals
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