❖ Calculating a Definite Integral Using Riemann Sums - Part 1 ❖

📚 Setting Up Definite Integrals with Riemann Sums: Step-by-Step Guide (Part 1) 🧮
In this video, I show you how to set up a definite integral using the method of Riemann sums. This is the foundational technique for finding areas beneath curves, and though it can be a bit tedious, it’s essential for understanding the concept behind integration.
🚀 What you’ll learn:
How to set up a definite integral using the Riemann Sum definition.
Detailed explanations of each step involved in creating the sum.
A historical perspective on why this method is fundamental to calculus.
In Part 2, I’ll actually compute the Riemann Sum to find the solution. While you may already know the faster, modern method to evaluate this integral, this tutorial focuses on the core concept and gives you a deep understanding of how integration was originally approached.
This video is perfect for anyone learning calculus, preparing for exams, or those interested in understanding the deeper fundamentals of integration. Don’t worry, I’ll also show you the quicker way to do this in future videos!
👍 Don’t forget to like, subscribe, and hit the notification bell for more math tutorials! 🔔✨
Watch now to master setting up definite integrals using Riemann sums and grasp the basics of integration!
#RiemannSums #DefiniteIntegral #Calculus #MathTutorial #LearningMath #Mathematics #IntegralSetup #STEM #MathHelp #IntegrationFundamentals #APCalculus #EducationalContent #AreasUnderCurves #IntegrationConcepts #ReimannSum

[@patrickjmt]

Hi all! Please post comments, questions and anything else on your mind in the comment section! While I try to answer as many questions as I can, there are more questions than I can answer by myself so if you are someone who might be able to help, please do so! The community is so much better when everyone is involved and helping each other out. Our strength is our willingness to help each other! Also please feel free to like or dislike the video. Your feedback is always appreciated so that hopefully I can make my future videos the best possible! Thanks and happy studies!

[@michaelm6597]

with this video i finally understand this stuff thank you so much

[@Safwan.Hossain]

This man posted this video in 2009 and still willing to help out and answer comments a decade later. The 🐐

[@rylflush3427]

Ad -"are you really trying to learn math from videos published in 2006" Math doesn't change every year bro

[@benzhang9186]

Brady Ho No, I am trying to learn math from textbooks published in ancient greece

[@Abhinav-ss3th]

@@benzhang9186 my school doesnt bother to update their books so these videos make sense

[@Safwan.Hossain]

Cuz these are fundamentals! If they somehow changed, the entire mathematical world would be in chaos for better or worse. Math does change every year, but only the very advanced stuff that goes way over my head personally.

[@andrewmorgan3100]

new pre-school math: "Am I a joke to you"

[@Kingmodder]

It is always very interesting to see people on KZbin who are a lot better than professors at universities, such as Patrick.

[@patrickjmt]

+Kingmodder well, I used to teach at university

[@Kingmodder]

Haha, yes, and probably most of your subscribers would wish that you were the one to teach them (for instance me) ;).

[@prezzoelsavo]

i do this in high school :(

[@limyeeshin]

Some people who good at something not necessarily good at teaching

[@IslamicScholars130]

Kingmodder exactly..

[@JamieJamBottle]

I've literally spent the past 6 hours going through sources trying to understand the Riemann sum for my year 12 Maths assignment, calling up friends in uni and going through their text books, and this one simple video explained what I couldn't understand for 6 hours in 6 minutes. You, kind sir, are a life saver. Thank-you so much!!!

[@robrick9361]

I remember watching this back in 09 when I was in school. So glad this guy blew up.

[@patrickjmt]

if you want to see where the formulas come from watch the video about 'the definite integral - understanding the definition ' that is listed on the right.

[@mdabubakar6874]

where r those videos

[@Xilosphere]

Dem video r n deez channel.

[@jalipenorunner]

you > khanacademy tho.

[@marcusmiguel1850]

fr nigga

[@alsilver7780]

Mad W

[@windthorpe9628]

They're both good... they can coexist...

[@umindujayabandu8954]

why u trynna start some fake beef?

[@fawfuls]

Both of them are great, and they have their differences. I like Pat because he jumps straight into the problems, whereas Sal takes his time and explains the problem in detail and how that problem originated (which isn't a bad thing, but most of the time I prefer to just see how it's solved if I already understand why it's important). If I'm really struggling with a concept, then I go to Sal.

[@patrickjmt]

Hi all! Wanna help a KZbin education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

[@HEQT0R_]

thanks man i got a quiz tomorrow and i dont understand this but you are helping a lot

[@yanirabayardo]

You have been the biggest help in all my college math courses :)

[@JonathanSwinger]

"Don't talk to me or my sums ever again" - Riemann (1865)

[@EuropeanSpaceAgency1]

I can't believe that of all the videos about the Riemann sum on KZbin, or any mathematical problem, your videos always clear things up. Good job!

[@jeremygraham7976]

I love to get drunk af an watch these videos, for some reason i can go math so well while being high i love these videos. I love you Patrick

[@jeremygraham7976]

Well the grammar is so off in this that I'm not even going to erase it

[@AhmadAlghannam]

Thanks God to let me find this channel. Peace upon you

[@Lucerozeus]

I'm taking calc 1 and Patrick's videos are extremely helpful. so I definitely agree with you.

[@LiveOutLoud87]

you're my favorite math tutor on youtube, thank you thank you! Always very clear and helpful with your examples. YOU ROCK!

[@TheBjamin88]

Patrick, it is truly remarkable, how great of a teacher you are. Sometimes teachers are extremely smart, but can't sufficiently explain the material, but you are the best of those two worlds. I have been watching these videos since 8th grade when I didnt understand something in school, and these always cleared things up. you have really helped me throughout my high school math career, and you deserve a sincere thank you. keep up the good work.

[@londiwemagwanyana1999]

funny how this video was published 5yrs ago but it's still valid and useful...thank u patrick for the awesome job!!!

[@djvhixa5884]

After 12 years it is still helping many thanks alot

[@parmida1988]

How I wish you were my professor. You explain things like no one else and I understand everything so much better when I watch your videos. It's good to be smart but to be smart and be able to pass the material to someone else (what you do) is just wonderful! Thank you for your videos and time!

[@chloekimball536]

Thank you for the video sir. You managed to teach me more in these 6 minutes than my professor did in an hour.

[@quenmoomoo]

This video. You saved me. Thank you.

[@patrickjmt]

ha! - one of the best comments ever : )

[@claregriffith1487]

This was so so helpful, I had tried to understand sigma notation for riemann sums for so long and this is the only thing that helped. The best!

[@samanthajr4648]

There is a god and he wants me to pass my exam... thanks, Mr Maths man! :D

[@4thKyuubi]

SUBSCRIBED! I swear, I LOVE EVERYTHING about MATH EXCEPT FOR RIEMANN SUMS. Butttttt, Thank you for explaining gazillion times better than my instructor! I have a test next week and this is like the only lesson i am not getting until i saw your vids.

[@Angelamica7253]

Taking the Ap Calc AB exam tomorrow, this video is a great refresher, now I have nothing to worry about! Thanks for sharing your awesome teaching abilities one the internet!

[@muhleraymond3386]

I used to study with these vids during my 1st and 2nd year of Engineering Math, now I have graduated just came back because I am helping my sister haha! These have been helpful, Thank you Mr Patrick! 🙏🏿

[@superflex007]

I took a weekend to learn this , but then i forgot it again. After seeing this video , i now know it again. Thanks !

[@Skyglory59]

TEAM LAST MINUTE💀🤡

[@catsrule256]

Watching this in line for my exam

[@qwertymouse2010]

you made learning 6 chapters of integration in one night possible! You are my hero!

[@shananerd112]

Thank you so so much! I cannt explain how grateful I am that you take the time to make these videos, and that they are free too! Nevermind the expensive online tutors or khan academy, who's videos are half an hour for one sub-subject. I've been watching your videos since I was a sophmore in calc 1, and a clueless one at that! No matter how many times I've been in for help from my teachers, none of them could explain these concepts as well as you have! You're a remarkable teacher, Thank you :)

[@MopTopRock]

When do you HAVE to use riemmans sums? Will the question always directly ask for you to use the Riemmans sum techniuqe to find the definite integral?

[@fenice2189]

This video helped me a lot. My Calculus book didn't explain it too well and my teacher just kind of assigns problems to us and expects us to teach ourselves basically and this video (and part 2) helped me so much. Thanks a lot man!!

[@ChocolateMilk19]

Thank you! I’m struggling trying to understand this with my homework due tomorrow! Much appreciated!

[@ScottyPotty666]

So i was studying for my Calc 2 miidterm and i realized i had no idea how to do this because i slept through lecture one day. It is currently 1 18 Am and my test is in 8 hours. I now know this so well. You are my savior!

[@TheSquigster]

It made me giggle right before 4:50 you kinda paused like, "ok, lets do this". You save my life btw, thankyou!

[@mounampesiyadhe]

As I painfully cram for my AP Calc BC exam tomorrow, your videos are my sole source of comfort.

[@andreatandog3593]

I was 4 years old when this video was uploaded and I'm watching it now 🤣I think that this video was made for me🤣

[@carrie1297]

patrickJMT is my best friend, always helpful and never leaves me hanging

[@anthonypipkin5545]

AP Calculus test on Tuesday anyone?

[@gladysruiz763]

Anthony Pipkin yeppp ab

[@heresthadude1080]

Monday

[@thepenmen22]

This is hilarious man. I'm studying for my exam, and when I have a question, I search on youtube. And everytime, there is a suggestion with your name on it haha. You are so freaking awesome!! These videos help so much. I wish teachers explained stuff like this...

[@enndruu]

my calc final is in two days; your channel is going to save my life

[@enndruu]

subbed of course

[@acousticsimon94]

You got me through my first University unit, thanks heaps!

[@yausernametaken]

Plz keep making videos. It is so so much easier when you do it. Wish I had you for my teacher

[@patrickjmt]

the point can come from anywhere inside of the interval. the point is, as n goes to infinity, the intervals become so small, it is basically the same y value no matter where you pick the point to be.

[@TheBubble811]

You are amazing. I watch these videos and literally go "Ohhhhhh!" because it just seems to make sense when you explain it. Thank you so much for all of these! Thanks to you, I might pass a calc test for once!

[@HoUsEoFlAsTwOrDs]

I want you to know this video helped me so much. Thank you. I was having a lot of trouble with the manipulation, but this cleared it up. You are a wizard and a scholar.

[@JakeOfAllTrades17]

I can't believe how easy you made this for me to understand. Thanks so much patrickJMT!

[@patrickjmt]

@Slappydappy if you are approximating an integral, you can use any point in the interval. to prove that an integral can be calculated by antiderivatives, it turns out you can again use any point in the interval.

[@Palmar3s]

@patrickJMT yeah. you should get an award or something too!!!!! I'm totally nominating you for teacher of the year!

[@patrickjmt]

@Slappydappy it represents any point in the ' i-th ' interval. the idea is that you really do not have to use left endpoints or right endpoints in the definition; all that is needed is to take a point from each interval. the x-i notation is pointing that out.

[@thetroofteller]

Conceptual understanding of the basis of integration is one thing, but why would ANYONE do this voluntarily? CHRIST! I have to teach this soon so this video was a great help. Question though - can you use the almost formulaic approach you used here with those summation results and the riemann sum limit definition? That would be much easier to teach than going into all the BS I've seen in other riemann sum videos. Thanks for all your great work and please keep it up :)

[@BrucasIsBettaThanYou]

YOU HAVE LITERALLY SAVED ME FROM FAILING. Thank you (:

[@fremsley001]

You can also calculate definite integrals by drawing the function as a curve on a piece of grease-proof paper, filling the area underneath with a uniform layer of cream cheese, then carefully scraping it off and weighing it on kitchen scales, giving the answer in "Philadelphian equivalents." This is the way Riemann himself did it before he hit on the idea of his sums. (PS. Andrew Wyles used cough drops and teaspoons to prove Fermat's Last Theorem.)

[@patrickjmt]

@ronniemonnie well, label points on the number line: x_0, x_1, x_2, etc and find a formula for their values.

[@nehakasera6560]

Thankyou so much. I have an AP exam tomorrow and it helped me through a lot

[@Philllin]

i learn math from PATRICKJMT....and im in college. ! thanks alot!!!!

[@RKN786]

Thanks a lot! You explained it hundred times better than my professor. :)

[@Madvillian01]

Patrick, I once again want to say thank you for all your help. I got my younger brother watching your vids too so keep makin em! :)

[@RANDINTMCR3U3]

Now I understand...no computer programmer will use the Fundamental Theorem of Calculus to evaluate definite integrals...Riemann Sums are straightforward, only a couple of formulas. Summation formulae can be derived from matrices as well, thank you, PatrickJMT.

[@robertwarshot1608]

You, sir, are a life saver

[@elisaav]

Heyyy!!! I soooo love your writing!! love those 2 you make :) ooh, and thank you so much for this video, I have a test tomorrow and this sort of made me feel a little more confident about what I know :) you're great :D

[@rizvif]

Hi Patrick, Thanks so much for the videos. I have a tutoring business and I use your videos for reviews. I am a chemistry person who tutors some calculus and your videos have been of a lot of help!!

[@xivalzzz]

thanks man this is gonna help me on my finals

[@timothynolasco8391]

great video of this lesson... i've passed most of my exams just by watching videos from @patrickJMT thanks for the videos, it really helps A LOT! :)

[@kutalpdilber8815]

exactly what i am looking for, thanks man

[@patrickjmt]

thanks a bunch ben. i keep making these videos cause of comments like this :)

[@syedishtiyaqahmed5133]

May God bless you brother . My phase test in one week . Thanks from India

[@sexybeckygurl95]

your videos are the best!

[@wind_tempos]

your* Unfortunately he does not make English videos as well.

[@21norki21]

Danny Vasta look at you, you're so smart and condescending. how cute.

[@wind_tempos]

21norki21 There is nothing cute about what I said.

[@and1fer]

Danny Vasta you made my day lol

[@sexybeckygurl95]

Danny Vasta yea, thanks for the correction, i got a little too excited.

[@HeyWheresKel]

plus, if we do bad on the exam, that just means we'll be better prepared for calculus classes in college! the important thing is we try. thanks for the help patrick

[@Postmark3]

Thanks so much! My teacher skimmed over this and this helped me a lot.

[@muningsun3483]

This is really helpful! Especially the equation part. Thanks

[@mohfa1806]

Hello patrick , thank you for your videos , they are great.....i have a question pls : on 1:03 it says Xi*=a+i(delta)X.....Shouldnt that be Xi*=a+i(delta)X divided by 2 ?? according to your previous video.....Thank you again

[@fauzanadityaharsya6146]

patrickjmt have commented : the point can come from anywhere inside of the interval. the point is, as n goes to infinity, the intervals become so small, it is basically the same y value no matter where you pick the point to be. i was also confused at first, then i saw the comment

[@jonabruno8243]

I dont understand too much enghlish but you are so good teacher that i dont even need enghlish language!!. thank you now am lill bit smarter :D

[@patrickjmt]

@theSteakosaurus you wouldn't. but this concept of taking something 'complicated ' (like the area of a general region), approximating it with many simple things (rectangles, in this case), and then taking a limit. the MAGIC is that using an antiderivative does help us calculate these things.

[@patrickjmt]

my pleasure, come back any time

[@patrickjmt]

@HREL7 very welcome!

[@courtneykramer6602]

Its 2020 and this video just saved me from my professor. Thank you!!!! :)

[@ta-tahuang4904]

thank you patrick you are such a nice person

[@ChristieCallie]

I have a test on this today. Thanks for this great review!

[@superstarben37]

Woooah! I just got my mind blown! This is CRAZY!

[@romnickjanoras5815]

Your videos are really helping me a lot. Thank you! ❤️

[@sariffadataya313]

I missed one class and so I ended up here lol this saved me 😁

[@Pandahombre]

Patrick it is nice to see you are left handed, just as I am

[@thelightning90]

your videos are very helpful. thanks.

[@babolokimadome7285]

Thank you Patrick, very much

[@thoyo]

yesssss! the book made this seem impossible to understand. can't wait to finish this semester so i can actually go back and relearn everything without worrying about ad hoc learning for an exam

[@tupacsomething69]

you sir have single handedly replaced my lecturers, and i go to university of sydney

[@ColdPotatoFilms]

AP BC Exam tomorrow. Needless to say im watching all your videos

[@MetalConch]

I love how your ads say quite "scrounging youtube for math help," haha

[@siguerhakim4723]

so well explanation, thanks

[@Playstationshow]

You saved my life. Thank you good sir!

[@patrickjmt]

almost... it would be a + (delta x)*i (that is, the i is not distributed to the 'a'.... i assume you mean what i wrote though!)

[@resistnzisfutl]

You are awesome, thank you so much for taking the time to make these very helpful videos!

[@elementoxygen]

patrickJMT, thank you soooooo much (really it would be infinity long tons!!!) You are the finest, greatest, top, foremost, leading, preeminent, premier, prime, first, principal, supreme, of the highest quality, superlative, par excellence, unrivaled, without equal, unsurpassed, peerless, matchless, unparalleled, unbeaten, unbeatable, ultimate, incomparable, and perfect in my math life 8D

[@Narulopo]

You saved my life!!!! Much much much thanks!!!!!!

[@nanone1994]

oh ! thank God my curriculum has canceled this torture !! Although it's so long , I find it interesting ! it's hard to understand the full picture without these basics i guess... thanks anyway Patrick ..

[@pukamon1681]

My teacher doesn't foil it out when you plug in x(sub)i in f(x). She uses the idea of limits at infinity. Will you be able to show how to that instead? Thank you!

[@iamradsara]

thank you for posting this video- it really does help!

[@armeniarule]

Thanks man, God bless you. You should write a book!

[@bencasica4047]

thx, im trying to prove the area of the circle using this just for fun