Riemann integration pinched out points

[shell tendon]

Not able to find the expression for the L(f,P)

[long hatch]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[shell tendon]

Here, i attached the way my prof solves it, yes we have some symbols we made on our own

@magic folio if u can help id appreciate it

[glacial ravine]

Not sure where the 8x(j - 1) came from.
Note that 3Δx(j) = -2Δx(j) + 5Δx(j). The second term can be included in the sum, so the result won't contain Δx(j - 1).

[shell tendon]

no i get u and u make sense, i think i did that whole thing wrong, with the substiutiion too f Xn and Xn-1 and Xj and Xj-1

[glacial ravine]

Well, you tried to write out the deltas, but you probably made an arithmetic mistake along the way. Anyway, that's just not needed here.

[shell tendon]

yes ur right, regardless i think the deltas itself are wrong like the first statement is correct from when i expand it its going wrong

[glacial ravine]

Oh, actually, yeah. You didn't write the result of the sum correctly.

The result will still be 5 and a small remainder.

[shell tendon]

Hii I got the L(f,p) mistake, I think as u said it was arithmetic mistake

[glacial ravine]

So, what did you get?

[shell tendon]

Of course from here on out I gotta use characterization of infimum and proofs

[glacial ravine]

No, that's not right.

[shell tendon]

[glacial ravine]

Again, just do this.

[shell tendon]

My friend told that to me

[glacial ravine]

Ah, wait!

Sorry, the second to last line is correct.

You didn't expand the delta correctly.

Why not just leave it at 5 - 2Δx(j)?

[shell tendon]

No it’s correct right? That’s how we do it atleast in class

[shell tendon]

Yeah we can leave it at that too

Either ways same thing right

[glacial ravine]

Δx(j) = x(j) - x(j - 1), so 5 - 2Δx(j) = 5 - 2x(j) + 2x(j - 1).

You wrote minus instead of plus in the last term.

[glacial ravine]

Well, yes, but the limit is more clear if you don't.

[shell tendon]

Oh yeah

I swear my brain is fried

[glacial ravine]

After all, we're taking the limit as the partition diameter (the minimum Δx) goes to zero.

[shell tendon]

Trueeeee

And I’m expanding it and making it a problem for myself

[shell tendon]

Oooo can u rephrase that

[glacial ravine]

What do you mean?

The definition of Riemann integral is the limit of integral sum as the partition diameter goes to zero.

(as long as that limit exists, is finite and doesn't depend on the actual partition, of course)

Alternatively, if the lower and upper some have the same (finite) limit, then the function is Riemann integrable and the definite integral is equal to that limit.

[shell tendon]

Yses

[shell tendon]

Idk my Ma’am never said it that way or explained that well in depth

Thanks for additional stuff

Maybe it was obvious but I never listened

[shell tendon]

Yes this we use

For our proofs

[shell tendon]

Oh wait

We did this too

damn wow my recollection power

Anyways thank u so much @glacial ravine 🩷

+close

+close

[glacial ravine]

You're welcome!

[glacial ravine]

Seems like the bot isn't working properly now again, so don't worry about that.

[proud sapphire]

@shell tendon

:HelpIcon:| Help Reminder

Hello aashi.op, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.

[shell tendon]

+close

[proud sapphire]

Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

[proud sapphire]

Thank you for your feedback! LordDarpinger has been awarded 1 . They now have 1154 . They have 3 daily left for today.