Relating Definite Integrals and Summations...

Just this week I started a Data Structures and Algorithms at my alma mater, San Jose state. And in jsut this short week I have noticed that the ones ability to properly evaluate ones code will take you a long way. Moreover, the skills developed through practiing the analysis of algortihm can go far in other discplines and studies.

A firm understanding and comfortablity with summations is fundamental to propering analyzing ones algortihm. In fact, coming form an EE background, I have seen summations many of times. And I have always felt as though I understood the underlying princple and implications of them. However, with that being said, I want to refresh myself. More specifically, create a document that I can later refer too and add to if needed.

Prior to anything in this class, I have always thought of summations as discrete representations of analog intergals. However, when I naviely assumed in the first reading that the trival summation, $$ \sum_{i=1}^{n} i$$ was easily translated to continous integral as, $$ \int_1^n i di $$ I quicly realized I was wrong once I did the math. For, $$ \sum_{i=1}^{n} i = \frac{n(n+1)}{2},\text{ (1) and }\int_1^n i di = \frac{n^2-1}{2} \text{ (2)}$$

At that point I figured it wpuld be a great investment to really look how the definte-continous-integral relates back to discrete-summation

Defintion:

Much of what I read in preparation for this post can be found here. Thank you UC Davis.

According the resources linked above, the definition for a definite integral related to discrete summations is as follows, $$ \int_a^b f(x)dx = \lim_{x\to\infty} \sum_{i=1}^{\infty} f(c_{i})\cdot \Delta x_{i}\text{ (3)}$$ where, $$ \Delta x_{i} = \frac{b-a}{n},\text{ the length of step interval, (4)}$$ and $$ c_{i} = a + (\frac{b-a}{n})i,\text{ right-end point of sampling interval, (5)}$$

Objective:

  1. Solve one of the sample problems from the link with the (3)
  2. Then work backwards from (1) with (3) to get the equivalent continous-intergal representation (2)

Solve Example

I decided to solve problem 2. It had similar representation to continous equation in which I thought eq (1) would yeild. So i figured doing a similar example to my problem would offer more insight.

The UC Davis site contained solutions. So for an indepth solution, I suggest one vist thier site. Also, they are many more solved examples to practice.

Problem 2:

Use the limit definition of definite integral to evaluate $\int_0^1 (2x + 3) dx$

With $f(x) = 2x + 3$, $a=0$ and $b=1$, we get $$x = c_{i} = a + (\frac{b-a}{n})i = \frac{i}{n} $$ and, $$ \Delta x_{i} = \frac{b-a}{n} = \frac{1}{n}$$ combining these equations in the left-hand-side of eq (3) we obtain the following equation $$ \lim_{n\to\infty} \sum_{i=1}^n (\frac{2i}{n^2} + \frac{3}{n}) = \lim_{n\to\infty} ( \frac{2}{n^2} \sum_{i=1}^n i + \sum_{i=1}^n \frac{3}{n} ) $$ from here, one can go view the detailed solution at the UC Davis link. I merely present up until this point to show I step the problem and used the eqs (3), (4), (5)