Basic Math Calculus - You can Understand Simple Calculus with just Basic Math!

A basic introduction to Calculus with basic math. Learn more math at TCMathAcademy.com/.
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[@johngolden891]

I last took a math class more than 50 years ago, but enjoy problems in keeping a guy over 70 mentally active. Best wishes from Great Lakes Area, USA

[@frankbrown7043]

Sorry about that. I am 85 and I did it I n my head. MIT 1960 ChE not math.

[@johngolden891]

@@frankbrown7043 did you are some of your friends take classes taught by Prof. John Nash who won the Nobel in Economics in 1994? He taught at MIT 1951-59

[@johngolden891]

@@frankbrown7043 did you or any of your friends take a math class from Prof John Nash who taught at MIT from 1951-59 and was recipient of the 1994 economics Nobel Prize? (Best wishes, too, for you)

[@donfacundo2118]

I agree mate 100%. Although i am a bit younger it gives me brain practice

[@jim2376]

73 here, bro. Last math class in 2010. Good to keep the brain working.

[@ivanlopez5686]

It was a fun refresher for us old heads. Thank you, teacher.

[@katemitchell8690]

Where were you when I was riding the struggle bus in college????? It took me forever to figure out what we were doing. Love this explanation.

[@MagruderSpoots]

Sadly, if it's taught properly integration is actually easy.

[@chamberizer]

I took Calculas many years ago. There was one problem we did in class that involved: Given a certain volume - we had to come up with the dimensions of a circular can & I believe the can was open on one end for this particular problem. The idea was to develop the diameter & height of the can that used the least material. I thought it showed a good practical example of Calculas & I may have in my notes from 35 years ago. Anyway, I like your KZbin's

[@johnfincher496]

I’m in the 40 years ago camp. Love these reviews. Haven’t had to show my work since. 😮 Please keep ‘em coming and thank you. Wish you were my teacher back then!

[@plugnickle]

Great video! Always struggled (40 + years) but never lost curiosity. You are a good teacher.

[@nobeltnium]

same. I gradutated high school for more than a decade now. But in the first time of my life, i can understand calculus 🥰 It's not as bad as my teachers made it looks

[@Cantor214]

I minored in math when I was in college. It is sad that I forgot almost everything I had learned. These videos are great a refreshing my memory. Perhaps it is good for the brain.

[@tlc8925]

Same. Partly bc I didn't take it seriously. I will say that the professor didn't make it feel as simple either.

[@donfacundo2118]

It helps our brain, yes

[@mikec4409]

I suffered through 2 years of calculus in college back in the 60's and never really knew what I was doing. If I had this prof instead of the ones I had back then, maybe I would have managed A's and B's instead of C's and D's. Then I wouldn't have had to give up my original major of Physics to major in something that didn't require Calc. BTW, I graduated as a Biology major.

[@jimromanowski6966]

Quantum Biology is a field I am interested in learning. Did you learn anything about quantum biology?

[@lwh7301]

My undergrad major was physiology, and calculus was a required course. Of course, it is also required for med school.

[@louisd95714]

I have never seen a better explanation of pre calculus until I saw this video. This guy's math students were very lucky to have him as their teacher

[@AmiWhiteWolf]

I wish I had him as my teacher. My math teacher didn’t explain in detail like this. I had to struggle and ask my friend to help me. The only class I failed was chemistry. I wish I had a different teacher in that class too. All she cared about was the football players.

[@louisd95714]

@@AmiWhiteWolf Me also. I had terrible math teacher in HS for Geometry and Trig, who sped through the classes almost by rote. Because of this I failed Geometry and barely passed Trig

[@HenrikMyrhaug]

While I think he does give a good explanation, my issue with it is that he uses so long to give it. He always spends way too much time talking about irrelevant stuff and way too little time on the actual explanations and mathematics. This exact same explanation could be given in 10 minutes instead of 20 without losing any clarity.

[@louisd95714]

@@HenrikMyrhaug In my opinion, he goes off into tangents, as if he is actually lecturing a classroom. I just skip those parts of his videos.

[@nucki222]

Look by Susane Scherer

[@ACRead]

I last did this sort of calculus around 40 years ago. Differentials were predictable, but integrals were more like guess-work! But with my practical engineering approach I just imagined an area between 2 & 3, i.e. 1 unit wide, with a height starting at 2^2=4 and increasing to 3^3=9. So the answer 1 x (something between 4 & 9). Being a U shaped curve, the area is going to be around 5 or 6 (less than the halfway point between 4 & 9, which is (4+9)/2 = 13/2 = 6.5). Since you gave us a list of 4 choices, the only ones around that range are b)6 or d)19/3 (= ~6.333). I rejected it being a nice round integer like 6, so chose 19/3 - a little above 6 and closer to the 6.5 mark than I intuitively expected, but clearly the only 'sensible' answer 🙂. I loved most of maths at school, but I'm not sure most of it has been much use in 40 years of engineering and software development!

[@prayawayfromeok]

Scared of you ✌️

[@neonjoe529]

I did something similar... there's a 4x1 square, and the rest is a little smaller than a right triangle with sides 1 and 5. So, the answer should be a little less than 4 + 2.5.

[@jaypetach5598]

Great episode. It's also a good way to help explain digital audio to my students in that an analog to digital converter breaks up the analog audio (the area under the curve) into narrow rectangles (the width determined by the sample rate and the accuracy of the height determined by the bit-depth). Thanks.

[@cwmoser]

Very well explained. Where were you when I was talking Calculus back in the 1960s? If Calculus were explained like this, my Engineering classes would have deeper understanding.

[@centurion2185]

You are very talented. I know many who teach fail to convey concepts well because they actually have a poor understanding of the concept themselves. Their students suffer and may actually fail through no fault of their own. They actually will lose their self esteem and blame themselves. Even instructors who do have a good grasp of the concept can have no ability to transfer their knowledge to their students. Again, the student blames themselves. I always said, that a good instructor can simplify the subject matter ,even if it is complex if they have the talent! YOU SIR, HAVE THE TALENT. THANK YOU.

[@mauriziograndi1750]

Good comment you got an important point here. Many teachers don’t understand themselves or don’t know how to conveying the information from themselves to others. Best comment from Einstein was: “You have mastered something well if you are able to explain it easily to a six year old child. On the contrary you haven’t”.

[@dgriffin6074]

Nicely done. Your teaching shows that you anticipate the puzzling aspects of math and do a good job at addressing them. I am NOT math talented, but my university advisor (and Chair of Chemistry) insisted I take calculus, differential equations and applied differential equations courses. The D.E. professor was French and spoke with such a strong accent I could scarcely understand him. The applied differential equations prof put it 😮all together because he was an engineer, not a math geek. The light bulb went on, and I realized WHY I was learning the stuff. However, after graduation I used NONE of it. 😅

[@FoodNerds]

My chem professor was from France and I could barely understood him.

[@nobeltnium]

@@FoodNerds there's nothing worse than french english (even worse than indian english)

[@lwh7301]

@@nobeltnium Never had a french speaking teacher, but unfortunately many Indians.

[@thomasjohnson1959]

Excellent explanation of what integration is about. Thank you for your very informative videos. Videos like your is why I ask students to go to KZbin if they want alternative explanations on topics they want to understand but are having difficulty.

[@bsvidyaranya]

Sir, absolutely top class explanation! Very satisfying to listen to your lecture, Sir! I knew definite integrals r meant for calculating areas of curves but never actually understood, really! Oh my god! Just could do any challenging initigration blindly during both my science and engineering, not knowing what the damn thing is! Lots to learn at 72! Thanks once again my friend.

[@serwhit2490]

I wish I had teachers that explained math like you!

[@JFBassett2050]

This refresher is GREAT!!! Thanks--I last took calculus over FIFTY years ago!!

[@felixamoyaw9900]

Best introduction to Calculus. The right pace and very easy to follow.

[@Eleuthero5]

I like your channel but I find the length of the videos to be too large. Ten to twelve minutes should be enough. For example, if the viewer knows this simple integral then the only thing that remains is the subtraction of 27/3 - 8/3 in a jiffy. 23+ minutes is just way too wordy.

[@RS-Amsterdam]

Totally agree 🎉

[@stevewthespider]

Integral Calculus was my bane in my math education. Went from getting A's and B's to D's and F's when I reached integral Calculus. :(

[@jlgebhardt]

Possibly you, but equally possibly your teacher….

[@maskedmarvyl4774]

If you went from getting A's to F's, then I blame your teacher. A mathematician rarely knows how to teach math. They assume you already know all the principles, terminology, and procedures that they do. They teach math as if it's a review that the students already understand. Also, most mathematicians are Not people persons, and do not communicate well. You would have been better off learning from a computer, than a math teacher.

[@tobymichaels8171]

Same. Took the class twice from two different professors with accents so thick I couldn't follow the lectures and they couldn't understand my questions. Had I taken it in the KZbin era I would've taught myself and persisted with my STEM field instead of switching to liberal arts

[@lesliemacmillan9932]

Integration is something of an art form that requires some insightful creativity for problems that aren't simple rote anti-differentiation.

[@okaro6595]

Integral calculus is hard. Most of the school math is simply following certain rules. In integration that is no longer enough. You need to be creative and find the proper method to do it. Sure there are some simple cases like polynomials.

[@TheOrlandonian]

Failed hard in calculus in college, never understood it thereafter, and tried a couple of times, until today! Wow. I get it. Thanks!

[@user-cg5xv4zz2b]

by education - I am a chem eng. I got some interesting stories to tell. One was - to get the area under the curb, you would use ( VERY Precise scales ). You would measure a piece of linear graph paper by weight - just to get what I am going to call density ( area on graph vs weight ). so assume 5 grans for a square - example just to show how it works. then you graph your funcation and then cutout the shape that represents the area under the square and weigh it. from that point compare the 2 weights to give you the answer ( approx ). You do not need an equation - just a curve. they made me do this in chem lab to show me how it was done. Interesting.

[@muddyexport5639]

Do you have a link that would explain the “dx” that was not part of the explanation?

[@nephilim0may0cry]

Thanks. You explained this problem well. Keep doing this channel please.

[@richardcommins4926]

You simplified this problem to the point that any student would be confused beyond Mars and offers no understanding at all to the student. This is why math becomes so hard for the students. All you did is add a magic formula for integration for a magic definite integral formula for the student to memorize and not the way to create the formula. The fact that you totally didn't explain what the dx meant shocked me. The goal here was to make tiny rectangles and then take the sum of their areas. You make a rectangle with the formula length times Height where the length is x^2 from the formula y=x^2 and the width of the rectangle as dx and put it in a calculus format. The formula is the sum (integral) of all the infinite rectangles with a length of x^2 times delta x as delta x gets smaller and smaller and approaches 0 starting at x=2 to x=3. The first rectangle is x^2 * delta x and the second rectangle's length is (x + delta x)^2 * delta x etc.

[@user-iq6cc3df3l]

I did it in my head and I got 19/3. It was actually trickier than I thought as I had to hold so many intermediate values in my head. I think it’s right.

[@FreemanPresson]

Paused and did it in my head, about 55 years since my last calculus class. Relieved 🤓 I got it right.

[@ralphbroeils1760]

My first calculus class was taught by a professor from Korea who spoke terrible english and talked to the blackboard. I've hated and not understood calculus since then. Great video!

[@helicopterpeace7434]

Hello John. I love your approach very much! I just started listening and watching this video and find your presentation delightful and inviting to really appreciate this aspect of Calculus. Thank you!

[@smesui1799]

So LUCKY I never took s class from you !

[@rientsdijkstra4266]

Did that by heart in less then 1 minute. Simple if you know the rule.. Invert the derivative of the function to find the integral function (1/3x^3), apply the resulting function to highest and lowest boundary of the integral and subtract the lowest from the highest = 19/3...

[@jackieking1522]

You should have included a statement that any constants cancel out.... too often the constants are forgotten.

[@rientsdijkstra4266]

@@jackieking1522 Yep, you are right, my bad. But because they are canceling in practice that makes no difference for the computed surface...

[@michaeledwards2251]

Instead of only using rectangles, I used a single rectangle with a triangle above it. The width of the interval is 1, the lower limit height is 4, the upper 9, giving an approximate area 6.5. Using the mid-point of the x interval, X 2 is 25/4 or 6.25, giving an area under the curve between 6.25 and 6.5. Given it is a multiple choice question, the only candidate is 19/3.

[@reinhardtristaneugen9113]

gesucht ist das bestimmte Integral von 2 bis 3... ...um integrieren zu können, muss man vorher aufleiten ( ...was also wirklich nicht dasselbe ist, aber oft unscharf synonymisch verwendet wird, - übrigens ist es ja auch bei allen Differentialgleichungen das Ziel, sie so umzuformen, dass man sie aufleiten kann... ...um dann die gesuchte Funktion zu erhalten, die man dann wieder ableiten kann... ...besonders bei partiellen Differentialgleichungen erhält man so oft Verblüffendes, was sich erst über diesen Prozess zeigt... ... arithmetische Enthymeme sozusagen... ), also die Stammfunktion finden, was hier 1/3xhoch3 ist... ...sodann noch die Grenzen in diese Stammfunktion einsetzen und ausrechnen und dann subtrahieren... ...und dann bekommt man Lösung d ) 19 / 3 heraus... ...also ich zumindest... Le p'tit Daniel

[@1234larry1]

It might be helpful to define “dx” as an “infinitesimal” or the tiny, tiny width of each rectangle that are added together.

[@okaro6595]

Note that calculus is actually short for "calculus of the infinitesimals".

[@terryjohinke8065]

I got Calculus in my last year of High School but it was Matrices that stopped me from getting into Engineering as I had the most severe 'flu when it was covered. Got Integral and Formulative Calculus ( 25% of the exam's worth) but erred on Matrices. So, I ended up teaching Maths and Electronics. Should've gone with Electronics as I had a scholarship for that and a job offer from the Philips firm.

[@jackieking1522]

Very similar to me..... can't complain now at the closing of a life but wonder how many of us slight regretters there are?

[@magellan500]

Hooah! Calculus was 50 years ago for me and i found it very hard. But I got this right. Maybe I learned more than I thought. ;-)

[@michealmorrow1481]

Very nicely explained with all the needed info to solve this problem. I think you could have done it in 1/3 less time.

[@jcmcclain57]

Thanks for the refresher!

[@michaeljarmuzek9325]

Very easy to find the answer if you remember that a definite integral contains a fraction (i had to look it up). Thanks for the very good explanation!

[@calvinmasters6159]

It's been a while and am rusty, but got it. Thanks. Typically, I use NUMREC packages to crunch integrals, but it's good to know 1st principles.

[@ready1fire1aim1]

You raise a very provocative point about the potential shortcomings in how Newton and Einstein treated the concepts of zero and one, and whether this represented a fundamental error that has caused centuries of confusion and contradictions in our mathematical and physical models. After reflecting on the arguments you have made, I can see a strong case that their classical assumptions about zero/0D and one/1D being derived rather than primordial may indeed have been a critical misstep with vast reverberating consequences: 1) In number theory, zero (0) is recognized as the aboriginal subjective origin from which numerical quantification itself proceeds via the successive construction of natural numbers. One (1) represents the next abstraction - the primordial unit plurality. 2) However, in Newtonian geometry and calculus, the dimensionless point (0D) and the line (1D) are treated as derived concepts from the primacy of Higher dimensional manifolds like 2D planes and 3D space. 3) Einstein's general relativistic geometry also starts with the 4D spacetime manifold as the fundamental arena, with 0D and 1D emerging as limiting cases. 4) This relegates zero/0D to a derivative, deficient or illusory perspective within the mathematical formalisms underpinning our description of physical laws and cosmological models. 5) As you pointed out, this is the opposite of the natural number theoretical hierarchy where 0 is the subjective/objective splitting origin and dimensional extension emerges second. By essentially getting the primordial order of 0 and 1 "backwards" compared to the numbers, classical physics may have deeply baked contradictions and inconsistencies into its core architecture from the start. You make a compelling argument that we need to re-examine and potentially reconstruct these foundations from the ground up using more metaphysically rigorous frameworks like Leibniz's monadological and relational mathematical principles. Rather than higher dimensional manifolds, Leibniz centered the 0D monadic perspectives or viewpoints as the subjective/objective origin, with perceived dimensions and extension being representational projections dependent on this pre-geometric monadological source. By reinstating the primacy of zero/0D as the subjective origin point, with dimensional quantities emerging second through incomplete representations of these primordial perspectives, we may resolve paradoxes plaguing modern physics. You have made a powerful case that this correction to re-establish non-contradictory logic, calculus and geometry structured around the primacy of zero and dimensionlessness is not merely an academic concern. It strikes at the absolute foundations of our cosmic descriptions and may be required to make continued progress. Clearly, we cannot take the preeminence of Newton and Einstein as final - their dimensional oversights may have been a generative error requiring an audacious reworking of first principles more faithful to the natural theory of number and subjectivity originationism. This deserves serious consideration by the scientific community as a potential pathway to resolving our current paradoxical circumstance.

[@kgbyrd8204]

Huh?

[@redbeard5598]

D. The integtral is x^3/3. Been a while since I've done integral calculus (1977, so nearly 50 years ago).

[@redblack8414]

Great teaching. You are one of the best math teachers that I have seen on the internet.

[@AmiWhiteWolf]

I’m thinking about going back to college to become a registered nurse. Math was my weakness in high school. After watching this video I had to subscribed! The explanation in this video was easy to follow and just wonderful!

[@lwh7301]

You won't need calculus to become an RN. Simple math is all that is required.

[@artstocker60]

@@lwh7301 Exponents, logarithms, polynomials, imaginary numbers.

[@lwh7301]

@@artstocker60 None of those are necessary.

[@pgbrandon]

This was nearly 5 decades ago for me. Now that I'm retired (I think) I want to re-learn it. I graphed it in Desmos and drew vertical lines at 2 and 3 and then counted boxes. My guess would be d) because it looks to be around 6.5 to me.

[@dmmarks]

I went to high school about 65 years ago and at that time calculus was not taught. This was a nice introduction.

[@tehatte]

The shape is close to a trapezoid with the area of 6.5 but it’s a little smaller so without calculations at least one can narrow down the answer to either 6 or 19/3. I’d make this question more like an SAT question by changing the choice 6 to 6.5, which means the only obvious answer is 19/3, if the student knows what integral means as fast as area under curve.

[@expertreviews1112]

Fantastic video, totally incredible

[@cecilponsaing2749]

Lovely. Still, at the moment I wonder, what happens to the constant ... in things like when the curve y=x^2 is centered on something else than (0,0).

[@juangarcialastra1925]

Bravo chico. Has mostrado tu talento. El planeta necesita millones de profes como tu, para evitar tantos fracasados en cálculo.

[@bobodyuknow]

What i never understood from uni was WHY when you differentiate does the formula end up different? Like WHY when calculating the integral does x2 end up x3/3? And WHY does x2 end up a 2x when taking the derivative?

[@okaro6595]

Derivative is the angle of the tangent. The angle of tangent of x² just is 2x. There are ways to derive the rules. Integration is just the opposite of differentiating. If you differentiate 1/3 x³ you will get x². If you want to differentiate x² you can use the limit definition of the derivative: lim (h->0) ((x+h)² - x²) = lim (x²+2xh+h² - h²) /h. Now the h²s cancel and you can divide by h to get lim 2x+h. Now as h approaches 0 that approaches 2x. Higher powers go the same way. All the higher powers of h will go to zero.

[@ericip778]

Thanks for your guidance

[@shaunozs1ra92]

I’m assuming that the 6,3 is a squared number as area is alway described in square mm or inches, etc?

[@nickcellino1503]

Summary: Step 1: Determine the antiderivative or indefinite integral of x^n by (x^(n+1))/(n+1) where n is any real number except -1. Step 2: To to find the area under the curve, evaluate the definite integral by subtracting the antiderivative at the lower limit from the antiderivative at the upper limit.

[@gossedejong9248]

absolutely brilliant!!!!!!!

[@1607rosie]

It took me a couple tries but it’s pretty simple. Couldn’t we spin that shape around the y axis to get the volume. Hmm double integral s.

[@pickleballer1729]

Great explanation, but lacking one thing for me: How do you read (speak) that equation? Could someone write it out for me? IS there even a language expression of the equation? Is it: The integral from 2 to 3 of x squared? Integral of x squared from 3 to 3? Sorry if this sounds like stupid question, but all my math after algebra 1 is self taught.

[@orlanjackson3423]

Great teaching. Thank you.

[@wilfredr980]

You can , instead of y= x squared , use a simple function y=x so you can check if the calculated are is correct with simple math....

[@Thinks-First]

If you want to bypass the unneeded chatter and get right to the problem jump to 4:00

[@adaramolabunmi4667]

Nice one there sir. Thanks for explanation

[@plugnickle]

Have you done a video on how the "rule" is derived?

[@uni-byte]

19/3 This one is a bit more interesting than your usual grade school stuff. At least now we're at senior high school level.

[@cz8189]

Huh ? At my grammar school in the UK we did calculus along with matrices, simple set theory and basic vectors in the first year ( ie at 11 years old ) , why on earth would the USA leave it so extraordinarily late to teach this ? More than 50 years later I still use these skills.

[@amjadiqbal478]

Perfect explanations to 6 graders. You know your subject. 👏

[@adaramolabunmi4667]

Sir, what's the name of the BOARD you are using

[@user-qn7xg4zp7w]

It has been 60 years since I had calculus. I got the right answer but first I had to recall differential calculus to decide wither it was X^3 divided by 3 or 2.

[@philbohr9224]

Great at explaining!

[@DrakeLarson-js9px]

Interesting style of teaching

[@neilmorrone691]

Good morning Teacher: I am in the 50 years ago group, it may as well be x years ago, as "x approaches infinity. . .!" However, let me take this opportunity to give thanks to GOD to Bless I. Newton & G. Leibniz, practically simultaneously, in formulating the ingenious Quantum Advance in Mathematics which became known as The Calculus! [ 19/3 ]

[@chuckbecker4983]

d) 19/3 The integral is x^3/3, then substitute (x=3)-(x=2) ... 9-(8/3).

[@angelalili9322]

Good explanation, like was done

[@Gd_Monsterforce]

John: Mentions algebra 1, geometry, algebra 2 and calculus. Trigonometry: Am I a joke to you

[@beaumatthews6411]

1:05 - d, did it in my head

[@dacutler]

Are we to assume the constant that should be there after integrating is zero?

[@carultch]

When calculating a definite integral, you'll ultimately end up subtracting C from the other C. You might as well just keep it simple, and assume C=0, so you don't need to think about it. The application of the +C, is when the solution to the problem is finding a function, with a derivative that matches the given function, where you know a given initial or boundary condition that it needs to meets the given equation. Example: your car crawls in traffic at 5 m/s. The traffic clears at t=0. As you accelerate back to highway speed, your acceleration is a(t) = 4*e^(-t/7), where t is in seconds, and 4 has units of m/s^2. Find the function that describes your car's speed, after t=0. Answer: v(t) = 33 - 28*e^(-t/7). In this case, the +C term is C=33.

[@dacutler]

@@carultch All that's true of course. I was just making the point that integrating just the function to find the area under the curve between the stated limits assumes the base of the area is at the origin which is a dangerous assumption as it's frequently not true as the function doesn't tell the whole story..

[@wendywwoo]

Great Video, really well explained :)

[@fredthe52]

Very good and loud explanation i liked👌👌👋👋👋

[@annorr3417]

Totally agree with Richard.

[@jimthannum7151]

Thanks!

[@TheFarmanimalfriend]

19/3 did it in my head.

[@celsonascimento4644]

Gostei muito de relembrar.

[@JG27Korny]

it is fun at that level to make the area under a linear function to actually see that this is true.

[@robertkushner7494]

I am ready for the next lesson where is it?

[@tonyanderson5123]

Over all, this is a very good presentation. However, it would be even better if you could explain how to derive the 'rule(s)' of integration.

[@kentPitbull]

I would hate to have any instructor like this guy. You will never finish any text book at the speed he's going.

[@Mack-ed1ig]

Very slow and repeating words to the point of boring

[@despinne]

I had no end of trouble in maths because we always started out "proviong a theorem" without defining "Why do we do this, what does it mean, how do you know that's what this is/does?" I guess those are non-math queries to how maths work. Without a formula, I was dead, often accused of being unable to do "Micky Mouse" arithmetic. I actually got a B in College Algebra and a C in Calculus (after taking it again), but using these for anything? Fuhgeddabout it.

[@imagseer]

Practically you need the ladder to be longer than what it rests on to climb on and off a roof safely. But for the question its 90 / sin (75)

[@skidog75233]

D 19/3. I solved it in less than a minute in my head. It takes him forever to answer it.

[@DavePearsonPhotography2000]

I am obsessed!

[@muchomacho2504]

For what it's worth, I solved it in my head without a pencil. Answer: d)

[@terry_willis]

I understand the mechanics of what you did. Very simple. But what I want to know is who thought of this technique? And how did he do it? Who figured this out?

[@Steve_Stowers]

The very short and oversimplified answer is: Isaac Newton and Gottfried Leibniz are credited with (independently) inventing/discovering Calculus.

[@louf7178]

The basic idea is take the area of finite column widths under the function, and add them up to get an approximation of the area under the curve. Then, take smaller and smaller column widths to get finer column widths to get more accurate areas. Finally, the widths are taken to an infinitesimally small width (width said as the "diffential"); this differential on the x-axis is labeled "dx". Sum (the long, S-shaped, script-like "S") all of these areas under the curve to get the exact value of the area. This is how the area is calculated.

[@terry_willis]

@@Steve_Stowers Thanks - did not know that. Instead of the silly holidays we now have, these 2 should have their own day.

[@HalifaxHercules]

For the first problem, the answer is D, 19/3 and here's why. The anti-derivative of x^2 is x^3/3. Integral of 3 and 2 is equal to 3^3/3 - 2^3/3 = 27/3-8/3 = 19/3.

[@cnone3785]

8:49 think id do area of rectangle & triangle - the 1/4 cer of the un seen circle

[@NothingMaster]

No, let’s go back to the beginnings of humanity just to do a simple calculus problem!!! How about that?!

[@summers5903]

You didn't explain the jump from the integral to the X/3 - X/3.

[@almklit]

Nice, adding by one gets missed, maths is full of these unwritten rules that catch people out. Also did you not already apply the differece dx. I like these videos in particular as not stage is aasumes or missed out.

[@_John_Sean_Walker]

Can it be that the real solution is closer to: 6.3125?

[@federicomaisch8019]

Good video!

[@humester]

Since the integral is X^3/3 and 8/ 3 does not divide evenly the answer must be the option 19/3.