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Introduction to integral calculus | Accumulation and Riemann sums | AP Calculus AB | Khan Academy

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The basic idea of Integral calculus is finding the area under a curve. To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areasÑcalculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus, as you will see in the upcoming videos.
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AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.
About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free, world-class education for anyone, anywhere. We believe learners of all ages should have unlimited access to free educational content they can master at their own pace. We use intelligent software, deep data analytics and intuitive user interfaces to help students and teachers around the world. Our resources cover preschool through early college education, including math, biology, chemistry, physics, economics, finance, history, grammar and more. We offer free personalized SAT test prep in partnership with the test developer, the College Board. Khan Academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. For more information, visit www.khanacademy.org, join us on Facebook or follow us on Twitter at @khanacademy. And remember, you can learn anything.
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Опубликовано:

22 авг 2024

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Комментарии : 145

@randomimmigrant348 6 лет назад

Khan Academy videos + Paul's online notes (Paul Dawkins I think) are a killer combination when learning maths/calculus. I went from a 51% (50% is pass at my university) for first year maths to 89% in 3rd year advanced calculus. I strongly recommend maths students to use these as learning supplements.

@PraveenKumar-kh5ns 6 лет назад

Random Immigrant what Is the 2nd channels exact name and does he summarise the important points of the chapter as his name suggests.

@randomimmigrant348 6 лет назад

@@PraveenKumar-kh5ns hi, If you're talking about Paul's online notes, it's not a channel, it's a website with high quality PDF's that explain concepts in detail and full examples.

@BhaiBhai-oz6pv 5 лет назад

1۔111ب2ب2ب2

@mashu3151 5 лет назад

Thanks for the tip !

@wojciechbrzychcyk5955 4 года назад

I was very bad student in first school, almost completelly absent in high school. After more than 20 years at the age of 42 I have started to study Computing, thanks to Sal I achieved distinction in math 100%, while having undiagnosed strong adhd- inattention. Khan Academy is the future of learning. Thank you Sal and other people that helping people to educate themselfs. P.S. Sorry for my grammar if it is bad, I never took english class.

@raeyan4244 2 года назад

Thats great! Im happy for you 😁👍🏻

@RacoonMedia 9 месяцев назад

I just started a bachelor degree in industrial engineering. Heavy on math. Never had that kind of math In my previous school. At 47 let’s see how far I can push my knowledge

@fatimaalaa2659 Год назад

I always get chills when I listen to Mr.Sal explaining the subject. He really takes you through a journey where you not only learn how to do the calculations, but also _understand_ the wonder of our complex & interrelated world. It's truly awe-inspiring. Thank you very much Mr.Sal

@gert-janroodehal7368 4 года назад

Told me EXACTLY what I needed to know. As opposed to any other source I could find. Thank you so much!

@erickjoshuanebreja6154 7 лет назад

Tried to skip from mathematical induction to integral calculus just to see how far math has become. What on earth is happening here....

@nivlanait 5 лет назад

xD in our school this is what some freshman in high school are learning

@BlueGuitarMusic 4 месяца назад

Go take a peak at topology or tensor calculus ;)

@lborate3543 7 лет назад

I love khan academy!! 3blue 1 brown has a great playlist on integrals also!

@trame8928 3 года назад

It's "too" not "also".😉

@scottyg222 5 лет назад

Learned how to count to 4 when I was two years old. Been earning a tremendous income as a professional musician for the past 45 years. Every once in awhile, I play a waltz...in which case, I only have to count to 3. 😁

@user-fr9ec4xq1e 4 года назад

keksksksk

@gagandeep5927 3 года назад

lol

@coolteateacool905 3 года назад

Nice comment, but why in this video?

@scottyg222 3 года назад

@@coolteateacool905 😐 Uh, because I have no friggin’ clue what he’s talking about. I was just being funny.

@brownpolyestershirt4701 7 лет назад

Found this channel on through my current professor. You explain things better than most math profs I've ever had. Thanks so much for helping me meet my goals.

@gregorybattis9588 5 лет назад

Yall explain things better in a shorter period of time than all of my previous teachers most brilliant moments combined.

@4amcuriosity162 6 лет назад

Just wanna say i never comment on these things but thank you especially for being the best most fluent youtube tutor page online. So helpfull. The background,color scheme, explanation all is on point

@andrejmilanovic6637 4 года назад

I apsolutely love you guys, went from medical high school to engineering college with little knowledge of calculus. You helped me with both integrals and derivatives so thank you ! Hope I pass my Math course ! :D

@azzteke 2 года назад

absolutely

@QuintonThai-wl8ue 2 месяца назад

6 years later and this is still godly. You rock brother

@jossieyan7480 4 года назад

why am i paying so much when youtube can explain what my prof did in 30 mins in 4 mins

@starRushi 3 года назад

because how are private corporations going benefit off you otherwise?

@nicholascarr6251 3 года назад

Because you are paying for the diploma not for the education

@meercreate 3 года назад

They know the paper is so important to your ability to earn money, so they gouge you accordingly, but since they're semi public and are subject to state auditing, they hire their buddies as "chancellor's of diversity" or something like that to justify tuition increases.

@dhruw9935 3 года назад

For the certificate

@Borzoi86 10 месяцев назад

I escaped high school without any exposure to calculus and, as an English major in college, escaped it again. But my curiosity never ceased . . . now I can learn without the pressure of a grade!

@kvOdratui 4 месяца назад

Thank you! Very easy to understand, now I also know about integrals! Thank you very much, hope you enjoy your day!

@MLG-gx9sp 6 лет назад

gotta love integral calculus :P thanks for the video

@killjason7425 6 лет назад

this is amazing. you have just saved me days to understand what an integral is. Thanks a lot and may god bless you

@cr7neymar908 Год назад

been with khan academy since algebra 1

@lukekennedy4970 6 лет назад

3:30-3:40 sounds so trippy😂

@sergiogonzalez1724 4 года назад

Luke Kennedy lmao what?! 🤣

@asia1652 26 дней назад

def my favorite part and the only thing I understood throughout this whole video

@tpc2000-v6p 6 лет назад

college algebra with trig is the furthest i want to go

@taasm1641 3 года назад

im in yr 8 and i learnt integral calculus from this guy

@chautran1827 6 месяцев назад

Thanks to video like this, I have a better understanding of math

@PianoRootsMusic 5 лет назад

so apparently I need integral calculus for my Physics IA (IB kids you know what's up) so now I'm trying to learn all this 6 days before the IA is due

@iprocrastinating8411 5 лет назад

I think I’m getting ahead of myself in trying to understand this.

@thiagokeizo 3 года назад

if you get derivatives first, then it becomes something beautiful I say

@Extorc 3 года назад

You are the Grand Saviour My Friend May God give like , what could you possibly need More knowledge MORE

@paigecollyns8285 7 лет назад

Hi I watch you in class

@thatturtleoverthere 7 лет назад

What kind of drawing tablet do you Sal use for the videos? Is it a cintiq?

@ayoubad5392 2 года назад

do you know wish software he uses ?

@AbyieAhmedam 2 года назад

u r the best teacher ...chemistry, math etc

@silentintegrals9104 2 года назад

yes!!!

@yebinji1800 11 месяцев назад

Youre explain is very good. i forgat what mean the integral calculus and when use it. but now i can remember

@PinkRosePurpleMauve1 7 лет назад

Bless Khan Academy

@monish4501 6 лет назад

thank you sir i can't understand the other videos your vedio is nice

@l1m19 5 лет назад

Thanks mr Khan

@NakSrea84 3 года назад

Teacher!,your video is amazing I love this subject and I will teach my student follow from you Thanks teacher for teaching video!

@abdulbari1971 4 года назад

thy is the maths god

@hamstershorts1699 2 года назад

Thanks for helping me teach caculus

@Prism_JTN 4 года назад

This was AWESOME!!!

@urmilakumari3706 7 лет назад

thank you Sir.

@DurgaDevi-uv4eo 6 лет назад

wow thankyou

@qualquan 3 месяца назад

Integration is much more than finding areas. Basically its about finding the WHOLE (W) given the fancy name Integral (I) from its Individual (not average) components (ICs) given the fancy name the "derivative". So finding the W fom its ICs is the same as finding the Integral (I) from its derivative. Now if the ICs or derivatives are mini lines the W or I will be the entire length as in finding the Length of a Curve. If the ICs or derivatives are mini areas the W or I will be the Whole area (under the curve). If the ICs or derivatives are mini volumes then the W or I will be the whole volume. If the ICs are mini Forces the the W or I will be the Whole Force and so on. Nothing special about AREAS at all.

@happyjulien6275 4 месяца назад

THank you

@YorangeJuice 3 года назад

I love 3b1b, but his videos on what an integral is come off as very convoluted to me; this ~5 minute explanation is much more clear. Thank you

@greenspringvalley 2 года назад

Why would you want to find the area under the curve? Is it just to calculate volume of real world things, or is it related to the graph?

@phenomenalphysics3548 5 лет назад

Please help the Khan academy app isn't working!!! Edit: its working now thanks

@lightningspark212 4 года назад

"Our journey of integral calculus" has been non-sensical so far

@thilokasurantha2266 2 года назад

Thank You Very Much ...

@guliyevshahriyar Год назад

This sound behind the videos, thanks a ton, you have taught very most concepts in fundamentals of mathematics best in the world.

@brallantp.2812 3 года назад

Calculus is just a mathematician that looked at literature and said, “I could do that.”

@henrybofoi8470 3 года назад

So happy for your video explaination 😍😍😍

@user-nf9xc7ww7m 4 года назад

Thank you, but what's the answer? My boss just wants the number and unit of measure. 28m²?

@songsforstatus2023 3 года назад

Clear explanation

@ummulsalmah4200 6 лет назад

Basically integrals is just better approximation of an area beneath a curve

@felixchien1664 4 года назад

NO! It's not quite a "better approximation"...it's suppose to be the "exact" or analytical area beneath the curve. It's the same as the derivative which gets the "instantaneous speed" rather than getting a good approximation. That's the power of calculus! By using the concept of limits, we can get the exact area or exact slope of any curve.

@No_BS_policy 3 года назад

@@felixchien1664 no not exactly. Limits are mere boundaries that a certain continuous function will keep on approaching, but will never arrive at. It will just keep on approaching infinitely. So there's no such thing as 'exact' area under a curve or an 'instantaneous' rate of change. We'll never get exact and instanteneous values using limits knowing its inherent numeric limitation.

@felixchien1664 3 года назад

@@No_BS_policy hmm, not sure if I agree actually. The concept of limits is that there is a definite number as something approaches a number (or concept like infinity). You shouldn't look at the limit of "n approaches infinity" as a "limitation" or "approximation". For example, the "limit of 1/n as n approaches infinity" is precisely 0. It's not that it approaches 0; it IS zero. A limit is a number. "It is not a process, nor an approximation, nor in any way imprecise."

@Mr-ep2qi 2 года назад

amazing thank you

@RubbinRobbin 5 лет назад

In autocad i would just draw any wonky curve and it would automatically know the area.

@skeletonsoldier7 Год назад

the real question is would it you help in examns

@march11stoneytony 11 месяцев назад

Incredible

@artsbeing4624 2 года назад

Thank you

@kevinlong9132 5 лет назад

the word you most used is infinistesinefily small

@angelxd7019 5 лет назад

No se supone que los rectángulos llegan tocado la función justamente en el punto medio del ancho de cada triángulo y así vez dejando un área libre y otra dentro para dar lugar a una compensación de área?

@remiremsar5946 Месяц назад

This is so f*cking cool.

@gavin4981 Год назад

This have at you.!!

@Bel...QWERTY1 9 месяцев назад

Could you please tell me how to know that integral of 1over x is ln x ... how they could know this in the first time ... what did they use series .. limits ..?! Is there any example how to exctract the rules from its origin not only to memorise them

@VikramSingh-up5ib 7 лет назад

Hey everybody can someone point me to where I can start learning calculus 1? Would that be limits?

@VikramSingh-up5ib 7 лет назад

+ADHD God thanks man, is there anything else in calculus 1 I should study as well?

@VikramSingh-up5ib 7 лет назад

+Ardian thanks man, I'll keep all that info in mind, appreciate it!

@dodiloi 7 лет назад

I have a Déjà vu

@naliniawasthi8886 6 лет назад

Physics

@naveedishtiaque 5 лет назад

ive just been this place before! HIGHER ON THE STREET

@asthavirmani5781 2 года назад

Ahhhhh thank you sm ily bye

@Yizee 7 лет назад

Is this real? I feel like you people write these strange and random letters to make us dumb people look even dumber. This is all craziness to me.

@pandiem 7 лет назад

Calculus and math is a ladder to be conquered. It's basically like learning the language of logic. You're not dumb for not understanding it. It can make sense to any dumb person if they learn it from the ground up. There ain't no shortcuts.

@zombiegamingtv9242 6 лет назад

What is that E like thing pls tell me pls

@Xaminn 6 лет назад

Check out this link: www.columbia.edu/itc/sipa/math/summation.html

@drespringfield8173 5 лет назад

It's called Sigma. Khan Academy has more videos over Sums & Series to help.

@sajkdnsaklfhnj 5 лет назад

Love you

@spacejunky4380 Год назад

Isn't it crazy who two opposite forces; that n gets infinitely large and I gets infinitely large have a symbiotic relationship? Litterally a paradox gives the most accurate summation you can never reach 😂 I'm incredibly grateful for Kahn Academy. I'm studying math in my free time before I pay for math courses. I love how simple they break down complicated ideas like integration. Derivates are neat and intuitive. Now integrals are short-of intuitive but to abstract still

@trollincharlie5408 5 лет назад

isthis where prople earn degrees from khan acaddmy

@mazeedom Год назад

Deu saudades de meu tempo de faculdade.

@videogamerstar1 7 лет назад

First thanks

@stephiie.89 2 месяца назад

The law says make me look like a joke and I'll leave ya feeling like a fool

@snorgisborg2 6 лет назад

I'm scared

@mikaoliveri 3 месяца назад

Why is a down there and b up there? What is dx? What's going on?? :(

@TheEyeist 2 месяца назад

Before starting integration you should go through the derivatives first they're connected you'll understand what dx is once you learn the derivatives

@stephiie.89 2 месяца назад

*doing shhhh sign*

@bikashsubedi2222 4 года назад

Any1 here after new vsauce video??

@spheralegg9609 Год назад

Indian kids learning this at 8th grade

@stephiie.89 2 месяца назад

Lol I fought the law once and the law won

@jah_141 18 дней назад

🗿notation

@stephiie.89 2 месяца назад

💙❤

@stephiie.89 2 месяца назад

💙

@user-qf8vi2gf9h 9 месяцев назад

G...

@pavlerakic3951 3 года назад

Can someone explain what do a and b define? are they like i and n?

@YorangeJuice 3 года назад

a and b are the lower and upper bounds respectively of the entire region you’re trying to find the area of on the x axis, they have nothing to do with i and n. n refers to the nth piece that you want to find the area of, and i is just used in the summation to make it clear what exactly you’re doing ^(If you need a better explanation of what i is in this case lmk because that explanation admittedly isn’t very clear)

@jaycarmona 2 года назад

Bruh what?

@govindanrajagopalan9904 6 лет назад

do not get it

@dsdsspp7130 6 лет назад

first start with limits and derivative

@user-jo7tr1rx8s Год назад

So now tell me, WHAT IS THE INTERGAL OF X!

@kenesufernandez1281 Год назад

❤♥️

@supernaturalstrength Год назад

God will help you. Get supernatural help. God is always on your team. God wants to be a part of your day. God wants to be a part of your schedule. Make room to spend time with Jesus.