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# Cant get a grip on integration. watch

1. So I started an integration module and I feel quite lost. It is the basics of integration and is basically showing how to define an integral in terms of area under a graph (definite) etc. I feel like I can do the questions set which just involve dissecting a given interval and taking the endpoint of all the intervals and adding them together and then take the limit as you get infinitly many strips. But I don't understand how we can take the right end point say and then use that, would the left end point give the same area if used to calculate it that way? Presumably it involves showing that both the lower end point and the upper end point have the same limit in the sum???

I have seen many proofs using sup and inf and delta epsilon proofs (again I think this may be proof of the limits) and I don't really understand them to be honest. Especially one proof that shows that the inf of the upper sum must be less or equal to the sup of the lower sum. Is this trying to show the limit is the same?

Also how can you define an indefinite integral as I only know how to define a definite integral using area, (is there any other way rather than saying it is just the opposite of differentiation).

I really like calculus and want to understand it better but it is confusing me very much.

I understand that this post is not all the clearest but maybe if someone can help me in a small way or link me some good resources that I can check out.

It doesnt help also that my lecturer is chinese and I find it very difficult to understand him as english is not both of our first languages haaaaaa.

Thank you guys!!!!
2. (Original post by poorform)
So I started an integration module and I feel quite lost. It is the basics of integration and is basically showing how to define an integral in terms of area under a graph (definite) etc. I feel like I can do the questions set which just involve dissecting a given interval and taking the endpoint of all the intervals and adding them together and then take the limit as you get infinitly many strips. But I don't understand how we can take the right end point say and then use that, would the left end point give the same area if used to calculate it that way? Presumably it involves showing that both the lower end point and the upper end point have the same limit in the sum???

I have seen many proofs using sup and inf and delta epsilon proofs (again I think this may be proof of the limits) and I don't really understand them to be honest. Especially one proof that shows that the inf of the upper sum must be less or equal to the sup of the lower sum. Is this trying to show the limit is the same?

Also how can you define an indefinite integral as I only know how to define a definite integral using area, (is there any other way rather than saying it is just the opposite of differentiation).

I really like calculus and want to understand it better but it is confusing me very much.

I understand that this post is not all the clearest but maybe if someone can help me in a small way or link me some good resources that I can check out.

It doesnt help also that my lecturer is chinese and I find it very difficult to understand him as english is not both of our first languages haaaaaa.

Thank you guys!!!!

My second course of analysis was just on calculus but unlike you I hate it ...

I plodded along through the course pulling my hair at times, and during exam leave I memorized (no word of a lie) around 50 theorems and lemmas.

I did get a comfortable first on that course, so plod along ...

(what University are you in?)
3. (Original post by poorform)
So I started an integration module and I feel quite lost. It is the basics of integration and is basically showing how to define an integral in terms of area under a graph (definite) etc. I feel like I can do the questions set which just involve dissecting a given interval and taking the endpoint of all the intervals and adding them together and then take the limit as you get infinitly many strips. But I don't understand how we can take the right end point say and then use that, would the left end point give the same area if used to calculate it that way? Presumably it involves showing that both the lower end point and the upper end point have the same limit in the sum???

I have seen many proofs using sup and inf and delta epsilon proofs (again I think this may be proof of the limits) and I don't really understand them to be honest. Especially one proof that shows that the inf of the upper sum must be less or equal to the sup of the lower sum. Is this trying to show the limit is the same?
This all depends on the exact details of how you've defined the integral. That exact definition will have the details required to make sure that you do get the same answer if you take the left end point, or the right end point (or the middle of each interval, for that matter).

It's very hard to give more info without knowing exactly how you've been taught an integral is defined.

Also how can you define an indefinite integral as I only know how to define a definite integral using area, (is there any other way rather than saying it is just the opposite of differentiation).
In analysis terms, an integral is essentially "area under a curve". But if you consider

this will behave very similarly to how you expect the "A-level style" indefinite integral to. (Note that the lower limit c is arbitrary - so this is where you get the arbit constant from on the LHS).
4. (Original post by DFranklin)
This all depends on the exact details of how you've defined the integral. That exact definition will have the details required to make sure that you do get the same answer if you take the left end point, or the right end point (or the middle of each interval, for that matter).

It's very hard to give more info without knowing exactly how you've been taught an integral is defined.

In analysis terms, an integral is essentially "area under a curve". But if you consider

this will behave very similarly to how you expect the "A-level style" indefinite integral to. (Note that the lower limit c is arbitrary - so this is where you get the arbit constant from on the LHS).
Ok thanks I am using the Riemann integral I believe I can't find the exact notes from my prof but I have been using this pdf as the definitions and theorems are the same with just slightly different notation.

https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf

Seems like good content to me, I spent about an hour looking at it yesterday and trying to get to grips with some of it and I'm definitely getting more comfortable I will see if anyone has anthing to add now I have given more info and try having a go at some exercises and prrofs myself later.

It seems like a lot of it relies on definitions of limits and epsilon delta proofs I believe.

5. I completely forgot to mention it was the Riemann integral apologies.
6. (Original post by poorform)
Ok thanks I am using the Riemann integral I believe I can't find the exact notes from my prof but I have been using this pdf as the definitions and theorems are the same with just slightly different notation.

https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf
OK, so this is a fairly standard definition involving taking the sup / inf of f on each interval.

But what I now don't understand is how this relates to:

(Original post by poorform)
I feel like I can do the questions set which just involve dissecting a given interval and taking the endpoint of all the subintervals and adding them together and then take the limit as you get infinitly many strips. But I don't understand how we can take the right end point say and then use that, would the left end point give the same area if used to calculate it that way?
Because this doesn't seem to mention sup / inf at all. I strongly suspect you are just evaluating f at a point in each subinterval. In which case if f isn't actually Riemann integrable then it is possible that evaluating at a different point in each subinterval could give a different result.

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