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March 30th, 2019, 03:07 PM  #1 
Newbie Joined: Mar 2019 From: sterling heights, michigan Posts: 1 Thanks: 0  Check my understanding (line integrals) 20190330 (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 subintervals (see attachment picture) so now if you want to integrate over the region we have along the curve C, using our subintervals, 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 subinterval along the region on curve C, then multiply that by the arc length of the subinterval in which our point in located in, and add up all the values you get for different points in each subinterval, 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 subinterval 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. 
April 8th, 2019, 12:19 PM  #2 
Newbie Joined: Oct 2018 From: Arizona Posts: 4 Thanks: 0 
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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