Elementary Integration - First Principles?

In summary, integrals can be solved by hand by starting with the definition and generalizing the pattern.
  • #1
SigmaScheme
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Is there an first-principles equation for integration that can be explicitly solved like that for differentiation? - I'm trying to understand Integration intuitively. Thanks heaps.

I've tried to piece together one but can't quite solve it (for simple functions like f = x). Things cancel and disappear and I get obviously wrong results.[itex]\int^{x_{2}}_{x_{1}}F(x)dx[/itex]=lim[itex]_{n\rightarrow\infty}[/itex][itex]\sum^{n}_{i=0}[/itex]F(x[itex]_{1}[/itex] + i[itex]\Delta[/itex]x)[itex]\Delta[/itex]x in which [itex]\Delta[/itex]x=[itex]\frac{x_{2}-x_{1}}{n}[/itex]

I want to know how integration was derived and how it works, like I do with differentiation and limits.

Thanks for reading my post.
 
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  • #2
SigmaScheme said:
Is there an first-principles equation for integration that can be explicitly solved like that for differentiation? - I'm trying to understand Integration intuitively. Thanks heaps.

I've tried to piece together one but can't quite solve it (for simple functions like f = x). Things cancel and disappear and I get obviously wrong results.


[itex]\int^{x_{2}}_{x_{1}}F(x)dx[/itex]=lim[itex]_{n\rightarrow\infty}[/itex][itex]\sum^{n}_{i=0}[/itex]F(x[itex]_{1}[/itex] + i[itex]\Delta[/itex]x)[itex]\Delta[/itex]x in which [itex]\Delta[/itex]x=[itex]\frac{x_{2}-x_{1}}{n}[/itex]

I want to know how integration was derived and how it works, like I do with differentiation and limits.

Thanks for reading my post.

Your topic needs too much reading/writing for a response in this forum. I suggest you DO some reading on the subject. There are numerous books available, and lots of material available on-line. One free article that seems to deal exactly with your issues is in http://www.maths.uq.edu.au/~jab/qamttalkmay2002.pdf , which was written for beginning students. Make sure you read the _whole_ thing; don't just look at page 1 and say "that is not what I need".

RGV
 
  • #3
SigmaScheme said:
Is there an first-principles equation for integration that can be explicitly solved like that for differentiation? - I'm trying to understand Integration intuitively. Thanks heaps.

I've tried to piece together one but can't quite solve it (for simple functions like f = x). Things cancel and disappear and I get obviously wrong results.[itex]\int^{x_{2}}_{x_{1}}F(x)dx[/itex]=lim[itex]_{n\rightarrow\infty}[/itex][itex]\sum^{n}_{i=0}[/itex]F(x[itex]_{1}[/itex] + i[itex]\Delta[/itex]x)[itex]\Delta[/itex]x in which [itex]\Delta[/itex]x=[itex]\frac{x_{2}-x_{1}}{n}[/itex]

I want to know how integration was derived and how it works, like I do with differentiation and limits.

Thanks for reading my post.

You seem to have seen Riemann sums, but this is the basic theory.

http://en.wikipedia.org/wiki/Riemann_integral

The computation of the limit of the Riemann sums is not always straightforward or easy.

It's very instructive to work out the Riemann integral of

[itex]\int^{1}_{0}x^2\ dx[/itex]

from first principles. You'll see when you do it that it's related to a well-known formula from discrete mathematics.

Then try [itex]\int^{1}_{0}x^3\ dx[/itex]

and see how far you can generalize the pattern.

Doing these by hand directly from the definition of the Riemann integral is an extremely edifying and also entertaining exercise. There's actually a mathematical punchline in there ... a little discovery that's the payoff for doing this by hand.
 
  • #4
Thanks!
 

FAQ: Elementary Integration - First Principles?

1. What is elementary integration?

Elementary integration is a mathematical process of finding the integral of a function using the basic integration techniques such as substitution, integration by parts, and partial fractions.

2. What is meant by "first principles" in elementary integration?

"First principles" in elementary integration refer to the fundamental concepts and techniques used to find the integral of a function, without relying on any predefined formulas or rules.

3. How do you find the integral of a function using first principles?

The integral of a function can be found using first principles by dividing the function into smaller parts, approximating each part with a rectangle, and summing up the areas of all the rectangles.

4. What are the advantages of using first principles in elementary integration?

Using first principles in elementary integration allows for a deeper understanding of the concepts and techniques involved in finding the integral of a function. It also allows for more flexibility in solving complex integrals that cannot be solved using traditional methods.

5. Are there any limitations to using first principles in elementary integration?

While first principles can be used to solve most integrals, it can be a time-consuming and tedious process for more complex functions. In such cases, using predefined formulas and techniques may be more efficient.

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