My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
June 15th, 2010, 02:11 PM   #1
Member
 
Joined: Feb 2010

Posts: 53
Thanks: 0

Integral as continous summation vs. area computation??

Okay, so I understand that in the follwoing integral:



that stands for those infinitetismally small widths of rectangles, and this inturn gives us a continuous summation of the rectangles under f(x). But what if I wanted to use the integral to add up the values of a function at everypoint? Say I had a function which takes as input time and outputs the amount of snow I had removed from my driveway at that time. If I wanted to add up all the values of with a continuous summation, from when I started shoveling to 5 hours lates I would use this integral right:

.

Heres what I dont understand, and yet understand, and then dont understand again. Since I just want to add up the values, and not find the area, whats the need for the , or more appropriately the value of in the counterpart of the integral, but whats the need for it? Dont I just want to add up all the values from 0 to 5?

So then I try this and realize that I simply get more and more values as I divide the function into smaller and smaller pieces. At first I may be adding , but dividing it up more I may be adding , and even more !!

This approach will simply go to infinity right!? So I understand that the stops this from happening, and I also understand its geomtric interpretation as the width of infinitismally small rectangles. My issues comes with understanding what this ellusive means when we are dealing with a continuous summation of a function like my snow shoveling one ? Is it an averager? Does it act to divide the function into its average? I know this cant be true because their is a seperate definition for the average of a function right!? So what the hell is anyway!? And please refrain from saying "the width of the rectangles", "a really small number", or "the back side of the integration function" as an answer. Thanks so much in advance.
mfetch22 is offline  
 
June 15th, 2010, 03:55 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,806
Thanks: 716

Re: Integral as continous summation vs. area computation??

You say R(t) is the amount of snow shoveled at time t. If it is instantaneous it would be 0, so I presume you mean R(t) to be the amount shoveled in some interval. That interval is ?t.

I you mean R(t) to be a rate, then you need to put in the time unit.
mathman is offline  
June 15th, 2010, 04:36 PM   #3
Member
 
Joined: Feb 2010

Posts: 53
Thanks: 0

Re: Integral as continous summation vs. area computation??

Quote:
Originally Posted by mathman
You say R(t) is the amount of snow shoveled at time t. If it is instantaneous it would be 0, so I presume you mean R(t) to be the amount shoveled in some interval. That interval is ?t.

I you mean R(t) to be a rate, then you need to put in the time unit.
You are correct my apologies, yes I mean that R(t) stand for the amount of snow that has been shoveled at that period of time. So maybe at the end of first hour (i.e t= 1) I'd only sovled 3 units of snow, but by the 5th our (i.e. t=5) I may have shoveled 12 untis of snow. That kind of function. What does the amount of snow shoveled multiplied by the infinitismal time increment , what does this value represent? In mathematical terms, and in terms of the snow shoveled.
mfetch22 is offline  
June 15th, 2010, 08:53 PM   #4
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Integral as continous summation vs. area computation??

Quote:
Originally Posted by mfetch22
You are correct my apologies, yes I mean that R(t) stand for the amount of snow that has been shoveled at that period of time. So maybe at the end of first hour (i.e t= 1) I'd only sovled 3 units of snow, but by the 5th our (i.e. t=5) I may have shoveled 12 untis of snow.
So in that case you have , where r(t) is the rate of snow shoveling.

Quote:
Originally Posted by mfetch22
What does the amount of snow shoveled multiplied by the infinitismal time increment , what does this value represent? In mathematical terms, and in terms of the snow shoveled.
Let's suppose for a moment that you shoveled at a constant rate of 3 units of snow per hour for the first two hours, then slowed to a rate of 2 units of snow per hour afterward. It's easy to see that this gives 3 units of snow at time 1 and 12 units of snow at time 5.

A first approximation of the amount of snow you shoveled (if we didn't know that it was 12 units) would be to take a sample point in the range, say at time 2.5, and multiply that rate by the amount of time, 5 hours. That would give 2 units per hour * 5 hours = 10 units. A better approximation would split it in two parts centered around, say, time 1.25 and time 3.75, with rates 3 units per hour and 2 units per hour; this would give 3 units per hour * 2.5 hours + 2 units per hour * 2.5 hours = 12.5 units. Splitting into finer and finer intervals will eventually get close to the true answer, 12 units.

In these examples, Delta-t was 5 and 2.5, respectively. If you split the range from x = a to b into k parts, Delta-x will be (b - a)/k.
CRGreathouse is offline  
June 16th, 2010, 09:47 AM   #5
Member
 
Joined: Feb 2010

Posts: 53
Thanks: 0

Re: Integral as continous summation vs. area computation??

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by mfetch22
You are correct my apologies, yes I mean that R(t) stand for the amount of snow that has been shoveled at that period of time. So maybe at the end of first hour (i.e t= 1) I'd only sovled 3 units of snow, but by the 5th our (i.e. t=5) I may have shoveled 12 untis of snow.
So in that case you have , where r(t) is the rate of snow shoveling.

Quote:
Originally Posted by mfetch22
What does the amount of snow shoveled multiplied by the infinitismal time increment , what does this value represent? In mathematical terms, and in terms of the snow shoveled.
Let's suppose for a moment that you shoveled at a constant rate of 3 units of snow per hour for the first two hours, then slowed to a rate of 2 units of snow per hour afterward. It's easy to see that this gives 3 units of snow at time 1 and 12 units of snow at time 5.

A first approximation of the amount of snow you shoveled (if we didn't know that it was 12 units) would be to take a sample point in the range, say at time 2.5, and multiply that rate by the amount of time, 5 hours. That would give 2 units per hour * 5 hours = 10 units. A better approximation would split it in two parts centered around, say, time 1.25 and time 3.75, with rates 3 units per hour and 2 units per hour; this would give 3 units per hour * 2.5 hours + 2 units per hour * 2.5 hours = 12.5 units. Splitting into finer and finer intervals will eventually get close to the true answer, 12 units.

In these examples, Delta-t was 5 and 2.5, respectively. If you split the range from x = a to b into k parts, Delta-x will be (b - a)/k.
This helps, but lets say we just have some function and I want to know what:



stands for mathematically without referencing to anything geometric or anything about its being the opposite of the derivitive. What I mean is what is this value mathematically, what does it mean to the function in no geometric terms? Or is there only a geomtric definition? My whole point of this question is to better understand how one can use integrals to continuosly add up cetain things. One thing where I am confused is the line integral over a vector field, being:



I understand that this integral adds up all the vectors along this path of curve C, but wouldn't that just be:

???

I know it isnt, but I'm just confused as to where or what the . I'm fimmiliar with vectors and their operations, and the dot and cross product so thats not neccesarily the problem. Is the line integral over a vector field to be considered as adding up all the vectors along the curve continuously? Or is there a better interpretation?
mfetch22 is offline  
June 16th, 2010, 10:17 AM   #6
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Integral as continous summation vs. area computation??

Quote:
Originally Posted by mfetch22
This helps, but lets say we just have some function and I want to know what:



stands for mathematically without referencing to anything geometric or anything about its being the opposite of the derivitive.
, if you like. There are many equivalent definitions.

The geometric interpretation is just there to help explain where this formula comes from.
CRGreathouse is offline  
June 16th, 2010, 03:55 PM   #7
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Integral as continous summation vs. area computation??

CRGreathouse's definition (actually, an equivalent, but slightly more general looking) is the standard one you'll see. The (geometric) intuition for the formula is that the product is the area under the curve with small, uniform-width rectangles. As the number of rectangles goes to infinity (and hence, the width of each goes to 0), the value of that sum approaches the integral.

If you're still confused about the "dx" at the end, just think of this as denoting what function you're integrating with respect to. You can actually integrate f(x) with respect to g(x).



And the definition of the integral that CRG gave changes a bit... Modifying the exact formula CRG gave is difficult, but instead of the width of each rectangle, you use a "weighted width", whose weight is dependent on the values of g at the boundaries of the rectangle.
cknapp is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
area, computation, continous, integral, summation



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
swapping summation and integral signs fromage Calculus 3 January 28th, 2014 01:20 AM
Computation of an integral. ZardoZ Complex Analysis 0 May 4th, 2013 01:12 PM
Integral Computation. ZardoZ Real Analysis 10 August 28th, 2012 09:25 AM
Integral Computation! ZardoZ Real Analysis 3 September 8th, 2011 04:49 AM
Integral Computation. ZardoZ Calculus 5 November 12th, 2010 05:06 PM





Copyright © 2019 My Math Forum. All rights reserved.