My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
March 30th, 2019, 03:07 PM   #1
Joined: Mar 2019
From: sterling heights, michigan

Posts: 1
Thanks: 0

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.
mhrob is offline  
April 8th, 2019, 12:19 PM   #2
Walagaster's Avatar
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
Walagaster is offline  

  My Math Forum > College Math Forum > Calculus

check, integrals, line, understanding

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Line Integrals DianaD Calculus 1 May 23rd, 2017 03:31 AM
Rearranging ODE's like algebra - please check my understanding Magnitude Differential Equations 7 March 13th, 2016 06:39 AM
Please double check my understanding whh2 Differential Equations 6 March 2nd, 2016 02:52 AM
Can someone check these integrals please??? beckham Calculus 1 August 13th, 2014 01:01 PM
Find the integrals in Q. Check your answer by differentiatio r-soy Calculus 17 December 21st, 2010 08:20 AM

Copyright © 2019 My Math Forum. All rights reserved.