Definite integral as the limit of a Riemann sum | AP Calculus AB | Khan Academy

Definite integrals represent the exact area under a given curve, and Riemann sums are used to approximate those areas. However, if we take Riemann sums with infinite rectangles of infinitely small width (using limits), we get the exact area, i.e. the definite integral! Created by Sal Khan.

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.

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[hennywisetheclown]

Is he using a mouse or a pen to write this? because if he's using a computer mouse, he's a fucking boss

[goldminer]

Was confused about the difference between Riemann sum calculations and integral calculations for a bit, this vid cleared it right up. When we do Riemann sum problems, we use a small, finite amount of rectangles represented by n, calculate their areas, then add their areas together to find an approximated area under the curve, we know that the true area is somewhere between the area of the left endpoint and the area of the right endpoint. Whereas with an integral, we essentially eliminate the error completely by using Riemann sum to calculate the area under the curve as n (number of rectangles) goes to infinity, leading to a more accurate result.

[SebastianLopez-nhrr]

I just gained a deeper understanding of calculus in 4 minutes

[ryanolstad]

He sounds like Paul from Llamas with Hats

[vasukumar]

i can binge watch his videos for whole day.

[adrianaquispepoma]

thank you so much! Good video hope you can make more videos about the Riemann sum and
integral

[JH-uxre]

Much better than my textbook. I don’t understand why they use so many words to explain math. Only by the several lines on this screen I understood it better than reading my textbook and taking my calculus 1 class

[tiancichen]

he's everywhere when i open youtube for edu. eco and maths. he is the big boss

[BLXXOR]

So Billy, how do you find the area of a square? RIEMANN SUMS!!!

[longhorn]

The word 'lim' here means that you cannot tell the difference between real area and the integral. The difference between real area and definite integral might be 0 and might be > 0. But in the second case you still cannot give the number that is less than the definite integral. So, it is not just 'a good approximation'. It is the best.

[nick]

Supposed to create a loop in matlab that does this...so that's why I'm looking at this video

[mohenderthakur]

woh that's very good for understanding

[LanRous]

A definite integral is the real value of the area of a positive function or just a good approximation?

[vighneshramesh]

just use 1.5 speed and listen, it is much much better

[victorwhite]

when can we just eliminate the sigma sign when taking the derivative of an infinite sum. I mean, for example we have the sum from 1 to infinity of 1/n; and that is equal to the integral from 1 to infinity of 1/n (which is lnn) plus the Euler-Mascheroni constant. In that case I guess we can take its derivative by simply eliminating the sigma, so it will be the limit when n tends to infinity of 1/n. When can someone do that same thing with other infinite sums?

[peelysl]

I'm an incoming 7th grader and I find this very fascinating. (:













ps. I'm going into Alg2 and participate in AMC8

[Dan-gcke]

why is delta x = (b-a)/n?. I get how b-a is delta x, why is it being divided by n?

[HenggaoCai]

you have to take the limit as x approaches infinite of the sum to get rid of the spaces between the rectangles

[jitendragupta]

wt is diff betwn integral and riemann integral

[minimeowmania]

This would have helped hours ago before I took a test on this