Please Subscribe here, thank you!!! goo.gl/JQ8Nys Definite Integral Using Limit Definition. In this video we compute a definite integral using the limit definition.
Пікірлер: 111
@iqdubz20365 жыл бұрын
This man deserves a trilliom likes read the book makes no sense come over here it all makes sense. Thanks man
@TheMathSorcerer5 жыл бұрын
Thank you man!!!!!!!!!!!
@mikaylachristiansen66037 жыл бұрын
This video saved my life! Thank you! I had been struggling to find a source to walk me through every part.
@ZachHixsonTutorials7 жыл бұрын
at 7:26, wouldn't that multiply out to (72n(n+1))/(2n^2), and then simplify to (9(n+1))/n? 36 would make sense if there was no +1, but I don't get why that +1 is just vanishing. EDIT: NVM, if you separate it out so it becomes 72n/n + 72/n, simplify it to 72 + 72/n, and take the limit, 72/n -> 0, so it basically becomes 72 + 0, and that's why it vanishes.
@JayTechMaster6 жыл бұрын
you mean 36 not 72
@aeyyyheller96614 жыл бұрын
Oopsie doopsie
@BrettAShore Жыл бұрын
Bro thanks for saving me a comment, was about to ask the same thing. Great explanation btw!
@brycezinckgraf76645 жыл бұрын
Thank you! This was so clear and easy to follow along!
@TheMathSorcerer5 жыл бұрын
no problem glad it helped!!!
@angelmendez-rivera3512 жыл бұрын
I should note that, technically, the definition given in the video is not the definition of the integral. If f is a function on [a, b], then you can partition [a, b] into intervals [x(i), x(i + 1)] with x(0) = a and x(n + 1) = b, and you can tag this partition letting t(i) be an element of [x(i), x(i + 1)]. Then the Riemann sum over the tagged partition is the sum of f[t(i)]·[x(i + 1) - x(i)]. The mesh of a partition is given by max[x(i + 1) - x(i)]. The integral is equal to the limit as max[x(i + 1) - x(i)] -> 0 of the Riemann sums. Here, we have that f(x) = 4·x^2 is a function on [1, 4]. So the Riemann sums are the sums of 4·t(i)^2·[x(i + 1) - x(i)]. Now, we have that max[x(i + 1) - x(i)] -> 0, so it is quite natural to have x(i + 1) - x(i) = max[x(i + 1) - x(i)] - d(i), with d(i) -> 0, so x(j) - x(0) = j·max[x(i + 1) - x(i)] - Sum{0 =< i =< j - 1, d(i)}. Let t(i) = x(i) + s(i) = x(0) + max[x(i + 1) - x(i)]·i + s(i) - Sum{0 =< j =< i - 1, d(j)}, with s(i) -> 0. Hence 4·t(i)^2·[x(i + 1) - x(i)] = 4·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)} + max[x(i + 1) - x(i)]·i]^2·{max[x(i + 1) - x(i)] - d(i)} = 4·{[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]^2 + 2·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]·max[x(i + 1) - x(i)]·i + max[x(i + 1) - x(i)]^2·i^2}·{max[x(i + 1) - x(i)] - d(i)} = 4·max[x(i + 1) - x(i)]·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]^2 + 8·max[x(i + 1) - x(i)]^2·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]·i + 4·max[x(i + 1) - x(i)]^3·i^2 - 4·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]^2·d(i) - 8·max[x(i + 1) - x(i)]·[1 + s(i) - Sum{0 =< j =< i - 1, d(j)}]·d(i)·i - 4·max[x(i + 1) - x(i)]^2·d(i)·i^2. Since max[(i + 1) - x(i)] is asymptotically equivalent to K/n for some real K > 0, one can use repeated applications of Tannery's theorem to conclude the limit as max[x(i + 1) - x(i)] -> 0 of the sums of the above is equal to the same limit of the sums of 4·max[x(i + 1) - x(i)] + 8·max[x(i + 1) - x(i)]^2·i + 4·max[x(i + 1) - x(i)]^3·i^2. The sums are given by 4·(n + 1)·max[x(i + 1) - x(i)] + 8·max[x(i + 1) - x(i)]^2·(n^2 + n)/2 + 4·max[x(i + 1) - x(i)]^3·(n^3/3 + n^2/2 + n/6), and as we have max[x(i + 1) - x(i)] -> 0, we get 4·K + 4·K^2 + 4/3·K^3. For reference, we have, from the fundamental theorem of calculus, that the integral is equal to 4/3·(4^3 - 1^3) = 4·(4^2 + 4·1 + 1^2) = 4·(16 + 4 + 1) = 4·21 = 84. So we know 4/3·K^3 + 4·K^2 + 4·K - 84 = 0. This has only one real solution, K = 3. In fact, the way one would go about computing K is by having K = x(n + 1) - x(0) = b - a = 4 - 1 = 3, but proving that this is the case is tedious and very laborious in itself. This is why we do not use the definition of the integral to compute them, and instead, we prove the minimal integral theorems, which are easier to do than computing any given integral, and then use the theorems instead. Also, if someone is wondering how exactly is this more correct, this is because t(i) is not being taken to simply be equal to x(i) or x(i + 1), but is an arbitrary number in the enclosed interval. Otherwise, s(i) = 0 or s(i) = x(i + 1) - x(i) respectively; and x(i + 1) - x(i) is not constant, hence t(i) is not merely a first-degree polynomial function of i. Otherwise, d(i) = 0.
@wyattoquinn81975 жыл бұрын
Thank you my guy. this was needed
@maclovely56 жыл бұрын
this was amazng. I loved how simple and easy to follow you explain your reasoning and every step. Definetly my new favorite channel for help.
@terryrodgers956011 ай бұрын
I always come to your videos, thank you for empowering my dreams!
@meanmole32123 жыл бұрын
5:25 Math Sorcerer wearing sunglasses in local high school's math club while throwing hand gestures.
@KatHammock6 жыл бұрын
this was soooo helpful, thank you!!
@alexjones19358 жыл бұрын
This video really helped me. Thank you!
@landonjorgensen19487 жыл бұрын
Thank you, not much else out there that explains this as well!
@TheMathSorcerer7 жыл бұрын
wow awesome, I am soooo glad this helped someone!!!
@elmersalinas62006 жыл бұрын
This saved my life, thanks!
@Fubbymaster5 жыл бұрын
Very clear. Thanks!
@jamesburmeister16085 жыл бұрын
Absolutely amazing explanation sir
@jovanniochoa7024 жыл бұрын
I love this. Thank you. You have helped me a lot.
@TheMathSorcerer4 жыл бұрын
awesome glad it helped!!!
@netogaming40484 жыл бұрын
You really must be a math sorcerer because that was magical 🧙♂️ thank you 🙏🏼
@TheMathSorcerer4 жыл бұрын
Np😄
@brycemackenzie10545 жыл бұрын
near the end, you pull (n(n+1)/2) up and say you just have remembered it, what are the other higher power ones, like if it was say i^3 or i^4
@hannahjoyceliz58217 жыл бұрын
thank you! I have a test next week and this helps a lot! :)
@TheMathSorcerer7 жыл бұрын
glad it helped!!
@jessem2866 жыл бұрын
Bruh thanks so much it came in clutch for my calc ll test
@shriyagupta65143 жыл бұрын
my ap calc exam is tom and you saved me. thank you so so so much.
@hxac_ Жыл бұрын
Great 👍🏼
@rebeccashelton85868 жыл бұрын
Thank you so much!!!!
@ikekennedy97003 жыл бұрын
How did you just drop all of the variables at the end? If you to foil it out wouldn’t you get a bunch of n variables?
@jamesyeung32863 жыл бұрын
You still have the limit as n -> ∞
@connordejager84497 жыл бұрын
You are an amazing human being
@TheMathSorcerer7 жыл бұрын
lol thx
@ourtube42668 жыл бұрын
My calculus grade says keep up the good work sir.
@amirdh17858 жыл бұрын
Thank you sir. Very helpful :)
@fcbarcatv-ou4ww Жыл бұрын
thank you so much!
@shirquutacraft46824 жыл бұрын
the12/n where is exteraNcomeing form? is that where 1can be a form of anywhere is that goes to infinity.
@megaprime677 жыл бұрын
I was pulling my hair out and slamming my head against the wall I couldn't figure out how I was getting it wrong after doing everything "perfectly" thenI watch this video and you saved my life at 7:30 "here you have to be careful" thats exaclty where i messed up, such a stupid mistake, thanks!
@TheMathSorcerer7 жыл бұрын
LOL that's great, glad it helped:)
@voodooblue61622 жыл бұрын
Would you recommend understanding how the formulas are derived or where they came from? I don't plan on majoring in Mathematics but I may try Computer Science which is essentially applied math I guess. I can get by Calculus 1 just fine if I practice using all the formulas, but I feel like I have not learnt anything meaningful though my goal is to achieve a high GPA and graduate ASAP.
@liamwelsh55652 жыл бұрын
I would say no unless your asked to prove the limit definition or prove other things that require the limit definition.
@josephward95897 жыл бұрын
Sorry, but I don't quite get how the summations work around 6:45. I understand what a summation is but don't understand how you get your results. Please can you help?
@josephward95897 жыл бұрын
Thanks :)
@adam_g.4 жыл бұрын
@@josephward9589 figure it out?
@coreyw.89887 жыл бұрын
I actually have a question, when distributing n to the quantities in the parenthesis, why is n(n+1) not n^2 +n? That would then leave the the quantity with a n on the top wouldn't it?
@coreyw.89887 жыл бұрын
im referring to 7:16 in the video, and the last part of the question as well
@rvne39597 жыл бұрын
If you actually multiply or FOIL everything out and simplify, the N's in the numerator cancel out with the n^3 in the denominator and all the parts with n in the denominator become zero because of the limit. It ends up leaving the final answer. This guy should have shown that instead of half-assing it.
@VeritasEtAequitas3 жыл бұрын
@@rvne3959 halfassing? Should he explain how to multiply and divide too? How he added the final numbers to get 84?
@alvinlepik52658 жыл бұрын
Can one work this scenario backwards? Were we given a limit and then come up with the equivalent definite (or improper) integral representation?
@TheMathSorcerer8 жыл бұрын
+Alvin the Chipmunk Yup
@melodys91884 жыл бұрын
bruh i just had a heLLA awFul day and missed this lesson in math but yOU my mAGicaL frieND just saved my l i f e and allowed me to nOt talk to people to learn this. huzzah!!!
@TheMathSorcerer4 жыл бұрын
Lol!! Glad it helped man
@Zerospaceman6 жыл бұрын
At 4:50 where did the 1 after the summation symbol come from?
@Zerospaceman6 жыл бұрын
Nevermind i got it but this guy should explain better what he is doing other than just doing it.
@maximumsdeggane5947 жыл бұрын
Is there a trick for an instance in which we have i^3 for example when we had i^2 it was just that number times 2 divided by 6?
@TheMathSorcerer7 жыл бұрын
ya i just think of the formula for i,so n(n + 1)/2, then square it,so sum(i^3) = n^2(n + 1)^2/4, it's just n(n + 1)/2 squared,
@user-yp1rg2jr5z Жыл бұрын
Can you evaluate {1/(1+x) dx} by using limit definition?
@user-yy3xd4vl8s3 жыл бұрын
How does 72/n squared x n(n+1)\2 equal to 36
@ElleHartStrings7 жыл бұрын
OMG thank you I understand it now
@TheMathSorcerer7 жыл бұрын
yay!
@aissatoudieng66537 ай бұрын
i really don't understand the simplification at the end.. shouln't we have 12+ 72n/2 because if we mutilply n^2 cancels out but we still have an n
@magg7764 жыл бұрын
great job, very clear
@TheMathSorcerer4 жыл бұрын
thank you!
@SeaBear1615 жыл бұрын
It is sad that you're better at explaining calculus than my professor, but it is true. Thank you.
@thomaspatrickparker66486 жыл бұрын
ok so my [a,b] is 2,6 but my f(x) = 8, so there is no function to plug anything in to. I've been doing the steps and found deltaX = 4i/n therefore Ci = 2 + (4i)/n my textbook is telling me the answer is 32 but I don't understand how you would get 32 from f(x) = 8, despite everything else. someone pls respond soon.
@mariamalkas60903 жыл бұрын
why can't my prof be this good
@audrey1236 жыл бұрын
This saved my future math career
@user123SSL Жыл бұрын
thank you
@vincentmichael0897 жыл бұрын
the last part is l'hopital or what? *the limit part*, please help, thanks^^
@kenrickchung81764 жыл бұрын
Under what circumstance would this method would be useful?
@TheMathSorcerer4 жыл бұрын
Wow good question, I don't know actually, can't think of one at the moment. Normally you use the formulas if you can.
@capnhype4 жыл бұрын
This is a banger
@TheMathSorcerer4 жыл бұрын
Lol thx
@EmOtEeDy7 жыл бұрын
Thank you!!!!!!
@TheMathSorcerer7 жыл бұрын
np!
@joshuamulleady639 жыл бұрын
nice video. thank you
@TheMathSorcerer9 жыл бұрын
Joshua Mulleady Np, glad it helped!!
@jonathanmendez97084 жыл бұрын
thank you!
@TheMathSorcerer4 жыл бұрын
You are welcome!
@sawyerhunt11267 жыл бұрын
wow!! unreal vid
@TheMathSorcerer7 жыл бұрын
glad it helped!
@blyatman23703 жыл бұрын
I thank the Lord everyday for your existence. And I'm agnostic!
@TheMathSorcerer3 жыл бұрын
Hehe thank you!
@Ahmed-vs1ui3 жыл бұрын
Why isnt Ci just equal to b-a
@omali84373 жыл бұрын
How is it 108/2 in the minute7:30 ?....and thanks alot💙
@VeritasEtAequitas3 жыл бұрын
And this mess is why someone figured out a way to integrate instead, lol.
@TheMathSorcerer3 жыл бұрын
Haha yes
@ronivansilva64474 ай бұрын
Can we solve an indefinite integral using limit definition?
@samrobinson38534 жыл бұрын
Thanks
@TheMathSorcerer4 жыл бұрын
no problem!
@jixster15665 жыл бұрын
4:26 ... What?
@TheMathSorcerer9 жыл бұрын
@theaya29507 жыл бұрын
Tmrw is my exam. Trying to learn calculus in one night thanx.
@TheMathSorcerer7 жыл бұрын
:)
@kilian82503 жыл бұрын
Where is the limit definition?
@TheMathSorcerer3 жыл бұрын
oh it's the "limit definition" of a definite integral, in other words, integral: lim n->inf sum(f(c_i)deltax),
@debasish2954Ай бұрын
Are there two definition of definite integral...@@TheMathSorcerer
@rolmangonzalez88557 жыл бұрын
@TheMathSorcerer7 жыл бұрын
:)
@JaneDoe-je7nh8 жыл бұрын
@JaneDoe-je7nh8 жыл бұрын
You have no idea how helpful this video is!!!
@TheMathSorcerer8 жыл бұрын
+Jane Doe I am glad it helped!!!!!
@Historia_Teotl2 ай бұрын
oh no, using limit process sucks
@TheMathSorcerer2 ай бұрын
ya it's way more work, if you can do ONE problem though on your own though, it becomes way easier