Just a theory question on the Integral

[flyingpig]

Homework Statement

The formal definition of the integral is

$$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\Delta x$$

The Attempt at a Solution

My question is, why does i = 1? Why don't we start at i = 0?

 

[eczeno]

it is just a matter of convention. the definition you give is derived from splitting an area up into little rectangles and summing the area of them. the convention is to begin the labeling of those rectangles with 1. so, when we sum from 1 to n, we get n rectangles, instead of n+1. since we are letting n go to infinity the result would be the same.

one point of confusion might be the labeling of the points vs. the labeling of the intervals (the rectangles). i was recently turned on to an article that really helps clarify this kind of problem. as soon as i figure out how to post a link i will link it.

 

[LCKurtz]

Homework Statement

The formal definition of the integral is

$$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\Delta x$$

The Attempt at a Solution

My question is, why does i = 1? Why don't we start at i = 0?

The partition is usually labeled:

a = x₀<x₁...<x_n-1<x_n = b

There are n intervals and n+1 endpoints. You can label the sum from 1 to n or from 0 to n-1 depending on whether you are using the left or right end point subscript to count with, and label the Δx accordingly.

 

[eczeno]

http://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/"

here it is.

 

[flyingpig]

If it is too much to ask, can you draw me a picture...?

 

[eczeno]

i will try, but don't laugh!

imagine three fence posts, and label them 0, 1 , 2. like this:

| | |
0 1 2

these are the 'points' on the number line, there are three of them, but they only form two spans of fence.

| 1 | 2 |
0_ 1_ 2

for integration, we want to count the 'spans of fence' (rectangles), not the 'posts' (partition points), so to speak.

 

[LCKurtz]

If it is too much to ask, can you draw me a picture...?

When there is more than one person responding to your post, it is a good idea to use the quote button and quote the person you are addressing so they know you are replying to them.