Elementary Integration - First Principles?

[SigmaScheme]

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.\int^{x_{2}}_{x_{1}}F(x)dx=lim_{n\rightarrow\infty}\sum^{n}_{i=0}F(x_{1} + i\Deltax)\Deltax in which \Deltax=\frac{x_{2}-x_{1}}{n}

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

Thanks for reading my post.

 

[Ray Vickson]

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.


\int^{x_{2}}_{x_{1}}F(x)dx=lim_{n\rightarrow\infty}\sum^{n}_{i=0}F(x_{1} + i\Deltax)\Deltax in which \Deltax=\frac{x_{2}-x_{1}}{n}

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

 

[SteveL27]

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.\int^{x_{2}}_{x_{1}}F(x)dx=lim_{n\rightarrow\infty}\sum^{n}_{i=0}F(x_{1} + i\Deltax)\Deltax in which \Deltax=\frac{x_{2}-x_{1}}{n}

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

\int^{1}_{0}x^2\ dx

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

Then try \int^{1}_{0}x^3\ dx

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.

 

[SigmaScheme]

Thanks!