The Definite Integral - Understanding the Definition

[bhlue05]

Thank God I found your video... It's really helpful... If only I could just enroll myself to your tutorial sessions, that would make my calculus understanding problems easier. :DDD

Thank you so much! XD

[CallOFDutyMVP666]

I find it amazing that you can add up an infinite number of infinitesimal cross sections and get a finite value.

[MrHishmeister]

This is exactly why I dont go to class :D !!! Patrick rules !!!!

[andr00123]

Find area of ƒ(x)=x^3 between x=0 and x=1 ∆x = 1-0/n = 1/n and Xi=0+i(1/n) so, ∫(x^3)∂x = lim ∑ (Xi^3)∆x = lim ∑ (i/n)^3 (1/n) <---replaced Xi with i/n and ∆x with 1/n = lim ∑ (i^3/n^3)(1/n) = lim 1/(n^4) ∑ i^3 <--- 1/(n^4) taken out so sum can be put in terms of n = lim (1/(n^4))((n^2(n+1)^2)/4) = lim (n^2 + 2n + 1) / (4n^2) <--- skipped steps to simplify (basic algebra) = 1/4 Why 1/4? Notice how the top power on both parts of the fraction are equal. Old property of a limit in Cal I

[andr00123]

@andr00123 Well that was a mess... sorry guys :( hope you get it.

[iSlapHoesOnUtube]

im paying so much money for my college tuition yet i dont get anything my professor says. i watch your videos for free and i get it. why dont i just pay you? the education system is messed up

[patrickJMT]

@iSlapHoesOnUtube i accept checks and cash : )

[fistful0fst33l]

@iSlapHoesOnUtube stfu u comment whore. go dish out some money to this guy then or go do something about it. You should probably drop out of college.

[jokerarayko]

How do you determine n?

[andr00123]

@jokerarayko n is the number of separations, or delta x's. When you find the true area using the limit, n is approaching infinity. The trick to solving the Riemann sum is to take the value's "n" as a constant (so that it looks something like this example: 2/n∑i). After solving the sums, you will be left with "n's" in a fraction. The numerator and denominator should have equal powers of n. The second part of this comment will include an example

[MrOverdoseXL]

I love this! So clear. :)

[patrickJMT]

@MrOverdoseXL perfect! : )

[TearfulCar1]

I can't thank you enough!!!!!



[ThePierdevara]

needs more boxes!

[richindiankid]

Amazing!

[ExtremeVoltages]

beautiful



[tikiman99099]

You have taught me more in 9 minutes than my teacher did all semester, you are a saint. Keep it up buddy!

[valdas0]

Thanks. Is there any video explaining why that anti-derivative is equal to the area under the curve? This video only shows that the right hand-side is the area.

[Angelgrrl04]

Wow thank you very much. My teacher is as clear as a foggy day.

[TRlPZ]

This is where calculus loses me.

[coolguy110395]

Great video! Understood the intuition behind integration. Your explanation was clear too!

[alexwhb122]

If you could not calculate an anti derivative for some function, is it possible to plug it into the destination and get an exact area and not just an approximation for some region? or is it that if you cant calculate an anti-derivative for the function you cannot find a exact solution in the definition?

[mahdibek]

your explanation simplified the definiton as much as possible..... good job :D

[lyrradXOX]

thanks a bunch man ^^

[NRC993]

simpson's rule ha

[erickvyadigg]

an advertising for cotton came up during this vid.

my head went through the wall

[illuzion30]

@8:28: "Problem is, how do you actually calculate something like this?" Argh... I was hoping you would explain why the anti-derivative gives us the answer to this summation.

[Carlos64073]

íf you came to Brazil in south america you would earn a lot of money by teaching students at " ITA UNIVERSITY; USP UNIVERSITY ; UNICAMP UNIVERSITY . These ones are the best in Brazil. The first one "ITA' has the more difficult college entrance exame in south america !!!

[sevinciCamus]

Why does it happen that when I read from a textbook I never understand the concept and after videos such as yourself or Khan's it all becomes crystal clear. Youtube as a method of learning is superior to everything else. If it gets integrated in classrooms no child would have to be left behind|!

[nupsyyy]

how can u take the height of all the rectangles x i?

[jimahoyy]

my professor takes 2 hrs to explain this.. you did it in 1/100000000000000 of the time. THANK YOU

[samagee24601]

THANK YOU SOOOOOO MUCH.

[Salamero500]

seriously you should be getting some kind of nobel prize for teaching excellence, your public service should be recognized some day by all of your students you never met, thank you kindly.

[ohorok2]

that video must be for retarded =D

[IMxNOTxVIOLENT]

I just passed my final and got an A.. JUST becuase of your vids..

I'm now shipping you a Pagani Zonda R.. be ready to drive like a maniac. :)

[Gazzawey]

u r the best i ever seen teaching math and calc i even didnt attend any class of calculas and i pass thnx alot and as u say spread the world with ur knowledge

[hbstrbo]

I dont need university any more

[Schism337]

I think its funny how its such a long process then at the end, Take the antiderivative then plug in B, subtract when you plug in A, and theres you're answer. Gotta love calculus lol

[MultiMoe91]

how come im doing this in calculus 1? :O

[Billabong024]

This video is amazing except its sound quality :)

[ndathebest]

You are the best!! you helped me alot to understand calculus 2!!

[ndathebest]

You are the best!! you helped me alot to understang calculus 2!!

[Psyondrax1]

You are STUPID! Make it simple and say, Find the antiderivate of the function, substitute with the interval's HOLY SHIT Fag don't make it difficult!

[Andyterranbase1]

@Psyondrax1 idiots... this is calculus not chemistry

[druisi]

@Psyondrax1 You're a total douche.

[DEKAT7644]

@Psyondrax1 You are filty scum. Get your emo ass out of here...

[PlayboyHZ]

my brain hurts. 

[christophersangma]

Very neatly done, brilliant!

[kuanliu9911]

you honestly saved my life.. ! thank you so much

[backcountryjame]

thumbs up for left handed people!

[travelholic91]

I'm only in pre-cal at the moment, and this video still made sense to me. Great video.

[snarejaguar1]

would you find xi* by doing (a + deltax times i) ?

[patrickJMT]

@snarejaguar1 yep, that is exactly what you do. you that when finding the exact value of a definite integral using the definition.

[dresector1]

I like this !!

[mhmthassright]

<33 Don't need to read Appendix A anymore! :D

[markov2b1]

in your definition, youre delta x should be delta x sub i . and the delta sub x i , and then there is a mesh thingy

[d3mur3]

i love you duuuuudeeeeeeee :D thanks

[bladerunner12]

The last time I took a Calculus course was 3 semesters ago, before I changed majors. This video still managed to make more sense to me than anything I had heard in class.

Wish I had found these sooner :\

[XxMurasamexX]

what's dx?

[99centstore]

This bears an uncanny resemblance the comparison between to analog to digital media storage... The function itself is the analog wave and the integral is like the digital approximation. Woohoo being a musician and knowing methods of music storage actually paid off for once! lol

[sonicgold1991]

THANK You!!!! omg.... life saver EXAMs 2mo wish em luck

[technicolorfuses]

thankkkkkkk youuuu patrickjmt. i love you :)

[ThePicaGirl]

THANKS... Patrick!

[mikelamid]

You're my new teacher. Thumbs up for Patrick! :)

[therock343]

WOOOWW... 2 hr Confusiing Lecture vs. 9 min Clear and simple Video.. Omg.. I love you Man! All Homo! haha

[Pwntzyou]

@therock343 Dude, I know exactly what you mean...

[criicrii100]

otro zurdo!!!!!

[mrwfraser]

Good man, you guys, people giving free lesson on maths, history, plumbing, web design and so on are unsung heros. Helping people for free. Good.

[Charounson]

Patt, how do I find the definite integral from 0 to x of e^(-t^2) ???

[patrickJMT]

@Charounson are you sure you are not taking the derivative of an integral (is that the question?)

[patrickJMT]

@Charounson otherwise, you would have to use a series expansion

[sim590]

Nice video, it's easy to understand. But, one point has not been mentionned. This point is that when you make n-->infiniy, you should also precise that deltaX-->0 even if that is obvious only by the fact that you say that n-->infinity. I think that this has to be mentionned, only because this video is dedicated to people who don't know this definition and therefor they have to understand all of it correctly.

Except for this tiny point, everything was good!

[HusseydayProductions]

correct me if I'm wrong but I thought it should be the limit as the norm of P goes to 0...not n to infinity?

[catagon12345]

I have a test tomorrow, your vids are really helping me study

[felixjrz1985]

patrick my question is: how do we proved that the summation of the small area is actually the integral? something is missing in your video how to you link the summation to integral. can you pls make a video that show the detail that the summation of infinitisimal value leads to integral.

[MakeMeBadKornPorn]

you make things a lil more simple, thank you very very much

[charmartist]

Amazing, I learn more from these videos than my professor lol. Whenever he goes over a new subject, I just refer to your videos for a better explanation

[IGNsucks]

this is how math classes should be like, I mean fuck whenever I go to my cal class Im tired, annoyed cause my school is an hour away and just bored cause i cant converse or socialize so its like why the fuck am i here. this so better, wake up when im rested, go on youtube learn math, rewind whenever i dont get it, do some problems. I think this is how it will be in the future for math and science classes.

[CamiloSanchez1979]

This is incredible ... my old teacher couldn't get me to understand this concept in 4 years....4 years repeating the same subject that is!

[proffezur]

@CamiloSanchez1979 Just drop out man.

[warlocks4life]

Thank you so much. My professor did not explain it as clearly as you have! I don't feel lost now. Again, thank you!!

[salo69]

If you ever come to miami... please le tme know and i'll buy you a beer... or wine... or wtvr u want cuz you are my mesiah

[patrickJMT]

@salo69 ha! sounds good! 

[carlosfernandez1989]

This video was incredibly helpful. Your videos will help me get through calculus 2 this semester. thank you for taking the time to make this

[patrickJMT]

@carlosfernandez1989 no problem!

[agona10]

my hero!!

[remolvr]

This really helps, taking AP BC calculus this year at the same time as AP physics C. I needed prior knowledge in Calc for the physics test so here I am using my summer to pre-take the course. I agree with a previous comment, you are my hero.

[Kale3193]

you are my hero

[patrickJMT]

@Kale3193 thanks, but i am not hero material.

[emilieminh]

Thank you!

[farshadguitar]

thumbs up if you believe this guy deserves the nobel prize !

[NineMillimeterKO]

I'm now im 10th grade and my math teacher once told me you could calculate the area under a curve, so a week after that I was laying in bed and thinking about what she said , and then I came up with the idea that would take the average y multiplied by delta x , havent learned about limits tho but this video basicly showed me that my idea was just perfect , I'm really glad that I basicly came up with the idea of calculating it and we didn't learn at school yet :o

[OletramX]

why can't DELTA-X be putted outside the sum, at the right side of the equal sign?

[sneakertri]

thank u very much

[Zadamanim]

This is really an extention of Archemides's formula, except he didnt have limits, so he used averages to come really close, these rectangles, as you add more and more (dividing the curve into smaller delta X's) the number becomes more accurate to the true area, how ever you need infinate rectangles to get a 100% accurate number, thus he uses the limit as X i approaches infinity

[coolguystatus]

for the hight couldn't you use one of the endpoints because as n goes to infinity and delta x goes to zero dosn't it all come out the same

[Gannondalf]

i dont know much about you, cause ive only watched two of your vids... but are you some kind of math teacher? if not, you should definitely become a lecturer and a university... you're really good at this

[healer378]

thanks, I went to college and got degree and passed grad still professors couldn't make sense to me tis helped but still I need practice. Thank you so much.

[cabecacere]

finally got it ! thanks

[tokio2t]

I finally understand what that formula means, thanks a lot! your very clear and I appreciate all the effort you take to make your videos.

[ramod123]

ur the best!

[hmorv]

great explanation!

[4thKyuubi]

so is this also the interpretation of riemann sums? Is the definition and riemann sum the same thing?

[conkerzzz]

Good, but can u please comress and limit your audio?

[622572]

omg i know this from my 15

[AJ410]

thanx very much

[vinumji666]

Thank u...

U r Gr8.....

[FranklyLate]

PatrickJMT:

Excellent video. Lucid and in 9 minutes!

Everytime you share knowledge you grow the world a little more and make it a little safer and better to live in.

cheers

[nickschor]

Further confirmation that oftentimes the idea behind the exercise is more important than the exercise itself! Thank you again...I'm swearing off of crappy reality tv shows and am going over my old math texts just to see why we did all this in the first place.

[Xcut33]

omg thank you so much,,,,i didnt get the textbook at all

[SrpskaBL]

Thank you very much!

I don't speak english well, but the quintessence is - reached ;)

Greeting from Republic of Srpska!

[kathysoiio]

oh gosh ..you are so good!!

[theitre]

Thanks a lot man. Its helpful and simple.

[zambiesz]

thanks! great video!

[ramsey1948]

thanks for a good video

[mathiaslylo]

You are extremely good at explaining this very nicely! Could you make one for Differential Equations too, that explains it this nice as well? Your other differential equation videos didn't explain it well enough to me. I like the way you present it visually.

[deityh699]

wow.. you make everything sound so easy and logical.

[wushuloveyou]

OMG!! You sir make calculus learnable to the average shmo.

THANKYOU

[evinpolack]

You truly are the king man

[KYAPOIU]

Great. This is the best video, presentation, teaching, best delivery, I have ever seen in my life.

Thanks you so much.. I love maths. I am learning it again after 17 years of switching to a different field. Trying to recollect what I learnt.

[jayinater]

thank you for being awesome

[kelmopro]

i love you man

[turningheadfart]

Finally  it all makes sense now

[wda8806]

i wish this guy was my teacher

[apratt111]

true say

[bluchamp]

Thank you. Much easier to understand when it's taught correctly. I swear teachers make things harder than they need to be.

[failingcargo]

this reminds me of "Good Will Hunting".

[VaughnCampbell]

not only do i not get this, your voice cracking makes me wanna punch my screen

[the0last0hero]

that my friend...is because you are retarded.

drop out of calculus and go play volleyball fag

[skyfaze]

vaughcambell ur hilarious u idiotic retard dont blame it on patrickJMT

[rolanpinto]

patrickJMT for president

[fandanstan]

@rolanpinto If I lived in the States, I'd vote! lol

[hayesjw]

dont mean to kiss your ass but.....uve saved me alot of anguish...thank u

[VillanuevaDaniel]

you could teach in chinese and still be better than most teachers.

[dinithsfernando]

uuggghh why cant teachers teach like this???? kudos my man

[b1ueflames]

Thanks. Finally that equation makes sense when I look at it.

[tma579]

You are awesomeman lol.

[RunLikePhat]

Thanks a lot dude. It really did help me understand where everything came from.

[Gansmiller]

haha im doing this high.. great call

[trunker420]

Wow that was super clear and easy to understand! I am doing this stoned and I only had to watch it once!!! lol you rock man I'm seriously considering donating.

[babicz]

"Hope this helps" ???  It was amazingly perfect!

[Yu2Kal]

Yes I agree this was very well presented!

[mrvlhs]

I've got a question: A derivative is a rate of variation. So, the integral being an antiderivative, should be the oposite of a rate of variation, and that is an area? It blows my mind. Your explanation is good but you start from the premisse that an integral is an area... well and how do you prove that an integral is an area?

If you base a theory on something wrong, the whole theory is wrong.

I'm not saying you are wrong, I'm just curious :)

[coolnickq]

dude its area under the curve. think about it like this: if the x-axis was labelled time (sec) and the y-axis was velocity (ft/sec), finding the area of a rectangle in this plane would be (sec)x(ft/sec) . if you multiply the two, it becomes just seconds of time.

[Druidiee9271]

Just an FYI: (sec) x (ft/sec) would be (sec)(ft) / (sec), or just feet simplified. However, the antiderivative of velocity would be distance (in feet going with your example.) Therefore, while your logic was flawed, your theory was correct. It also shows mrvlhs how an integral can be considered as an area when the function lies solely above the X-axis.

[coolnickq]

true i messed up the units there

[skin18]

Thanks a lot :)

[Ifrit0s]

i would write lim dx->0 too

[SatoTM2]

ur summation-sign sucks xD

its not a pyramid on the top! not a lightning-sign for the shaft >>

but still a great vid thanx =D

[dustylee]

i have a question:

is the dx, which follows the integration sign, the derivative of the function?

the simbol is ignored so many times in text books and lectures but it lingers there, and our ap class just began integration so i have no idea what these symbols mean

[pokerguy17]

@dustylee

The dx is the same thing as a small chunk of the x axis. Think about it, you have a rectangle with length and width, with length being y or f(x) and the width being dx. So, to find the area just integrate several rectangles with length f(x) and width dx from a to b to get the area bounded by the graph f(x) and the x axis.

[pokerguy17]

@dustylee

function f(x)***

[dustylee]

thank you.

i also noticed dx in derivatives and i can relate this to the small chunk of the x axis because dy/dx is the slope of two points of extremely small difference.

it also helped me to think that instead of delta y/delta x which is the average slope, i think of dy/dx as the instantaneous slope.

[shellaxx]

you're really good at this ! thanks so much , keep them coming

[yabangulum]

thank u for posting these videos.. i reall appreaciated it..it helped me so much..

[Xoxaloacetate]

this guy is better than my calculus bc teacher

[Awsomedotcom]

i find it totally fucking ridiculous that i can learn more in this one 10 minute video than i can from going to math 252 lecture class for an hour four days a week

[classicmusic05]

lecture? an university lecture? wtf

i'm in grade 12 and we are doing this...

[C1u3z]

Yes, I'm doing this for my Math 1001 Class. Which is normally taken 2nd semestar of first year. Math 1000 > Math 1001 > Math 2000

[classicmusic05]

ok, now i feel like my teacher is teaching too much if this is university stuff :(

but then again, it never hurts to know it

[C1u3z]

It depends on the school. Some teach Calc, some teach Pre-calc, and some teach advanced calc.

[C1u3z]

Yes,  I agree.

[senor47]

true...he gets to the point and does it great

[Lunchsacks]

haha totally agree!

[degrotube]

@Awsomedotcom Most teachers are baddddd. Thank you internets.

[chanabnagar]

could you use some better microphone ?

[Lunchsacks]

suggestion: if you want him to use a better mic, send him a new one, or money to buy one.

[chanabnagar]

what is his address or bank account ? :)

[Lunchsacks]

lulz

[chanabnagar]

I think the problem is not with his mic rather with signal level. When his voice level (loudness) crosses a critical value there is distortion due to clipping of the signal.

[HOBODC23]

Thank god i found this. Exam tommorow

[FlashFox5]

thanks... you should take a profession in teaching in the U.K!!!

[vectorlost]

2:27 you seems very tired :P

Thx for the infos, I owe you a lot.

[larmisdor]

This is very helpful. You are amazing. Keep up the good education. Thanks alot Mr Pat.

[tincata90]

this really really helped. thnks man MR. Pat. u saved me ALOT of time

[kingfker]

cool man :D... the best man ;)