BASIC Math Calculus - AREA of a Triangle - Understand Simple Calculus with just Basic Math!

[@terry_willis]

Excellent explanation. This comparison of both a triangle and thingamagig shape made this one very clear. John gets an A+ and a smiley face.

[@DCOneFourSeven]

I've been revising my calculus lately (and having fun doing so), and it was reassuring that I was able to "get" this video with relative ease. Practice makes perfect.

[@genelowry5666]

Agreed

[@chrisdissanayake6979]

Actually, I had forgotten my Calculus. This was a great review! Thank you so much, Mr John 🙏🏽

[@AbalisObidah]

wow! you made mathematics seems super easy. keep up with the fantastic job!

[@ZillaYT]

Excellent presentation.

[@girmaybass68]

Took calculus in 1996 - very good refresher! Thank you!

[@siyungjung7162]

Thank your for your detailed and clear explanation.

[@MariaMarreiros-il6gv]

Excellent didactics!

[@RajSharmaKumar]

I have been trying to get the practical application of calculus for a long time. This simple and beautiful example clear the concept clearly. 💯✅💐🌷🌺🎋

[@BrenoStrato]

It has thousands of applications.

[@stevendebettencourt7651]

The area of the triangle is A = (1/2) * b * h Where A is the area of the triangle, b is the length of its base, and h is the length of its height. Since we have a right triangle here with the base and height already noted, all we need to do is plug in the values and do the arithmetic: A = (1/2) * 6 * 6 A = (1/2) * 36 A = 18 square units (remember that area is always measured in square units of some kind)

[@redblack8414]

The purpose of the video was to calculate the area of the triangle by using calculus in case you didn't notice.

[@stevendebettencourt7651]

Well then, for the heck of it, let me demonstrate why y = (x^2) / 2 is the antiderivative of y = x. We will do this by showing the derivative of y = (x^2) / 2 is y = x. When you get to Calculus, you will be introduced to the following formula: Limit (h->0) [f(x + h) - f(x)] / h Basically, all this is saying is that as a tiny amount h trends to 0, you approach a value that is the derivative of the function f(x). Plugging in f(x) = (x^2) / 2, we get the following: Limit (h->0) {[(x + h)^2] / 2 - (x^2) / 2} / h = Limit (h->0) [(x^2 + 2xh + h^2) / 2 - (x^2) / 2] / h = Limit (h->0) [(x^2 + 2xh + h^2 - x^2) / 2] / h The x^2 terms cancel out, leaving = Limit (h->0) [(2xh + h^2) / 2] / h Let’s factor out a 2 first = Limit (h->0) [2(xh + h^2 / 2) / 2] / h The 2 terms in the numerator cancel out = Limit (h->0) (xh + h^2 / 2) / h Now we factor out the h in the numerator = Limit (h->0) [h(x + h / 2)] / h We can now cancel the h terms in the numerator and denominator, leaving = Limit (h->0) x + h / 2 Since division by h is now gone, we can just plug in h = 0 to wrap this up = x + (0) / 2 = x + 0 = x So, we have shown that the derivative function of f(x) = x^2 / 2 is f’(x) = x (derivative functions are often noted with this notation, which is usually read as “f prime of x”). Additional note: As Mr. KZbin Math Man says at the end of the video, while Calculus can seem intimidating, it is critical in so many ways to our world, well outside the field of mathematics itself. If you go into more advanced sciences, you will find that practically all of them will use Calculus extensively. Engineers use Calculus in their calculations all the time (insert obligatory joke about approximations that would NEVER EVER be allowed on a math exam). While you may end high school math with Calculus (hopefully), it would be wrong to see it as an end, but rather as the entrance to a new journey into a much wider world.

[@WilliamFarrow-c2r]

I don’t understand Calculus at all, but even if I did 1/2 base*height is the way to solve this. No one in the real world would use Calculus. If the intent is to demonstrate Calculus, use a question the needs it. Occam’s Razor at work.

[@jeemjeem6190]

Am I missing something? The parabola is shifted up from 0 to 2. Wouldn't you have to add 2 to the formula to find the area? BTW I do appreciate your videos.

[@artgilkey1547]

I agree. It should be: F(×)= x^2+2. X^3/3 + 2x

[@29brendus]

@@artgilkey1547 yep that was bugging me alright!

[@mayaq8324]

That would be a two hours video

[@BMGBOX]

Yes...just add the area of the 6 by 2 rectangle to his result...

[@BMGBOX]

Sorry: 4 by 2 rectangle...

[@uweruebe8854]

Eine super Erklärung zur Einführung in die Analysis. Hier wird den Schülern die Angst vor integral/differezialrechnung genommen - ganz easy 👍🏼

[@ajayprakashkelotra]

Like your speaking, teaching value ❤ Thanks ❤

[@HansjeVdTillaert]

In an earlier video, I mentioned that you could speak a bit faster, but for this particular issue, I feel a slightly slower pace might have been helpful. You're doing amazing-keep up the great work on your channel! 😊

[@pattywaktdiy4175]

Sheesh!

[@profittaker6662]

Calculus made simple most best explanation . .

[@ARIGROSSMAN]

I knew the answer using both methods. I'm just appreciating your approach on how to teach. Simplicity. Of course coupled with Teaching the student needs to practice.

[@ravianandh3346]

Can you please help me in what way Calculus is applied in finance practically please, sir

[@MohammadSami-ul-lah]

Amazing and informative videos.

[@nandisaand5287]

5:14 Shouldn't the formula for this parabola be Y=X²+2?

[@lortiz6563]

Yes

[@djbeacon6895]

Fo Sho!

[@BrenoStrato]

If You know the displacement (2), you can use x² and add the displacement later as 2x. You'll end with x³/3 + 2x. Or you can use x² + 2 directly, ending with x³/3 + 2x + C. The results will be the same.

[@МаксЛисак-й3н]

WOW, this teacher is really professional!

[@johnmarchington3146]

Doing the calculus, the integration results in x²/2, so that is 18 - 0, or 18 square units and that agrees with the area of the triangle.

[@gopalsamykannan2964]

The way and answer is correct. But how can we select correct function 'xdx' ?

[@yemanehailesellassie751]

I am not new to "Integrals" but I enjoyed it. Small note: Once you solved the integral, instead of writing X^3/3 - X^3/3. It was better to write as X4^3/3 - X0^3/3. Because some students may ask, why are you replacing different numbers if both are X's.

[@russelllomando8460]

18 using alg. easy calc was a bit tricker. couldn't remember all the Function rules. too many decades of non-use. thanks for the explainer.

[@maths01n]

Great ❤

[@AegidiusREX]

Can anyone tell me the name of the software used in this video? I've been looking everywhere but can't find it. Thanks in advance for your help.

[@robertb8417]

In your example under the curve y=x^2, it should be y = x^2 + K (a constant, a number)

[@Ignoranceisbliss-i2e]

what...utter nonsense...

[@BrenoStrato]

It doesn't matter if you know the value of the displacement (k). You can add k later as k•x.

[@tomtke7351]

(1/2)x^2 + C | 0->6 =(1/2)(6^2)+C - {(1/2)(0^2)+C} =(1/2)(36) =18

[@JohnDoe-ok5wo]

Show us the proof of this rule for integration or how it is derived.

[@Astrobrant2]

Well, you'd get the area under the parabola if you used the correct equation for the curve. Meaning you would have to add the extra 8 squares that are between the x-axis and the y-intercept.

[@robsawalker]

At 4:40 in, shouldn't that be like y = x squared plus one (or two or something)? Wouldnt that affect the calculus? Sorry if its a dumb question I know nothing about calculus or algebra!

[@BrenoStrato]

Whatever. If you know the displacement (2), you can use x² and add the displacement later as 2x. You'll end with x³/3 + 2x. Or you can use x² + 2 directly, ending with x³/3 + 2x + C. The results will be the same.

[@darshansinghrahal2549]

Show the lower and upper limits on triangle diagram

[@CarlWinter-oy8uf]

So if a triangle has its top apex broken off--instead of 1/2 base x height --you use calculus to find the reduced area?

[@TO-ps2qq]

Thomas Jefferson taught himself calculus. It was called fluctions back then. He then used calculus and invented the plow, for farming. John Deere made that plow out of steel.

[@CarlWinter-oy8uf]

So. substituting in 4. times X cubed --over 3. equals ---?

[@theobjectivethinker64]

...and if you plug in different numbers along the integral x you will get the different areas from 0 - 6. Rise over run gives the gradient.

[@raya.pawley3563]

Thank you. Well, this keyboard doesn't give me the notation I want, so making do. A= [1/2(x^2)] at (A@x=6) - (A@x=0) A@6 = (1/2)(6^2) = 36/2 = 18 A@0 = (1/2)(0^2) = 0 18 - 0 = 18

[@audionmusic2787]

The area of a triangle is equal to half the area of the smallest quadrilateral which can encompass it.

[@BrenoStrato]

This assumption is wrong.

[@darshansinghrahal2549]

This function shows area under Curve but no Curve in answer diagram

[@coolbombe]

That is not the equation of the parabola; it's y=x2+2.

[@BrenoStrato]

Whatever. If you know the displacement (2), you can use x² and add the displacement later as 2x. You'll end with x³/3 + 2x. Or you can use x² + 2 directly, ending with x³/3 + 2x + C. The results will be the same. 🤖

[@jim2376]

FTC. (6^2)/2 - (0^2)/2 = 18 units^2.

[@verdisquo97_arts]

area of the shape is 21.33, (int=4^2x^dx) or 64 into 3. literally first time doing calculus

[@gierotsan7918]

Square triangle is very easy to find Area in second ... because it's a half of square or rectangle shape... just use this formal S= ab/2 ..

[@philsaspiezone]

18 by using the 1/2b*h and again by doing the integral and plugging the 6 into it from the top of the integral symbol and leaving me free to ingnore the zero because it would be 18-0 using the fundamental theorem. I haven't done calculus at college.

[@pauldow1648]

Why not work the calculus demonstration problem ?

[@labvirtualfree]

exercicio: calcule a area de x = 1 ate x = infinito, da curva y = 1/x .

[@pierrebringmans4274]

So how much is the area???

[@photashoptip704]

Pronunciation is fine

[@josephshaff5194]

Final minus Initial again ? (I'm headed back 1st BS Physics) (curves are my next thing) (studying them until things calm down on Campuses) (Calc I Completed 4.0. That was some work! lol! Daily)

[@Ananya_anoop]

1/2 [x`2 ] 6to o = 36/2 = 18 Sq unit

[@RPL1000]

What is y =? when the triangle base = 3 and height = 4.

[@BrenoStrato]

y=4x/3.

[@wheres_bears1378]

Why was X not squared in one example but then squared in another?

[@md.noorulkarim5542]

Equation should be Y = x2 + C Area = 1/3 x3 + cx

[@stephenvarghese3657]

I was totally confused to see the parabola is not touching the origin when y=x2

[@BrenoStrato]

Hi. In this case, you can use x² If you know the displacement (k, 2, or whatever). You just have to add this area later (as kx or 2x, or whatever).

[@aibymuzic]

I think your explanation is as complex as the problem

[@ylmazgumus1578]

I think the formula of parabole must be y=X²+2

[@josephshaff5194]

Yay ! 🤣

[@jasonmudgarde286]

Am I daft or is it just half of the square? Understand it's about teaching integration but...

[@BrenoStrato]

In this case, you are finding the area formula algebraically (as a function of x). This has some advantages, especially when it comes to more complex surface areas.

[@stevenhaas9622]

technically that function is Y= X^2 +2

[@jansmit4628]

When you say "each triangle", but I think you mean "each rectangular triangle". When the 90 degrees angle get more or less, the area will decrease down to zero.

[@martinhicks6020]

No. The area of a triangle is always 1/2(base x height). It does not need to be a right angled triangle.

[@jansmit4628]

@martinhicks6020 Only for a triangle with a 90 degrees angle. If you decrease the 90 degrees angle to let's say 30 degrees and draw a new line from the top to the base, you end up with 2 rectangle triangles on which we can use the A= 0.5 x base x height rule. With h = 6 x sin(30) = 3, the area for the first triangle becomes 0.5 x 5.2 x 3 = 7.8. After calculating the base from the other triangle, its about 1.2. So the total area becomes 7.8 + 1.2 = 9 which is not equal to 18. I think the misunderstanding between us in my opinion is that there is one particular triangle which can be name rectangular triangle (one 90 degree corner) all othe have sharp(less than 90 degrees) or blunt(more than 90 degrees). The 0.5 x B x H is only applicable on the particular one. Just like the fact that a square (4 equal sides) is a particular rectangle (2 sets of 2 equal sides).

[@pierrebringmans4274]

I guess it only for isosceles triangle??

[@nandisaand5287]

No, it works for ALL triangles. What he didn't explain all that well is that the slope of the hypotenuse for the triangle (rise/run) is 1, therefore the formula is Y=X. If it was a 3/4/5 🔺️, the slope would be 3/4, therefore the formula would be Y=3/4X, and the integral would be Y'=3/8X²+C.

[@pierrebringmans4274]

Thank you​@@nandisaand5287

[@nucki222]

good

[@bitajafari3120]

The formula for the curve you showed is y=x**2+2 and not y=x**2 Thanks.

[@argonwheatbelly637]

You're just trying to find the area under the "curve," y=x, where x in [0,6].

[@CarlWinter-oy8uf]

Do we have to memorise that stretched S formula like a parrot ?--S --4 --0- X squared DX---- what does it mean ?

[@sergiofernandes2408]

SHOULD Y NOT BE Y=X2+C

[@hearstzhang3881]

There should be a constant

[@josephshaff5194]

ME's will use GD&T involving True Position Tol Stack up stats will employ RMS or Monte Carlo Method. EE's head toward Physics in antenna Design. And energy in the Field surrounding the wire concept. If ya can prep 'em for that that's where it's going. For Engineers. Research no idea. 🤣 Northup pulled a Iron Blast door across a room increasing a Magnetic Experiment. In the NEWS. Would've like to have seen that f/ a safe distance! 😂

[@Nevermore3540]

1/2 x base x height 1/2 × 6 x 6 1/2 x 36 = 18

[@maxpolaris99]

isn't that curve x^2+2 not x^2

[@mbirega]

18 I used calculus

[@Paul-it1oz]

On the odd shaped area how was xsquared plus one determined. Before it was x to the 1+1. What determined x to be squared plus 1.

[@hearstzhang3881]

Y = x^2+c

[@wasimahmad-t6c]

6×6÷2=18area 6×6=36×2=72squrot 8.482/6/6

[@panlomito]

0 S 6 x dx = [½ x²] = ½ . 6² - ½ . 0² = 18 units² Nice... and now a rotational integral...

[@frankw.5751]

the parabola looks to be y = x squared + 2

[@audreydaleski1067]

28+c. +c or its wrong

[@hamdikurdi1277]

i thing y have not right formola

[@matthewaustin8237]

18

[@CarlWinter-oy8uf]

That 45 degree angle line you call Y = X ----do you mean "length " of X is same as length of Y ? I am confused ! (can see only ONE 45 degree line )-------Maybe Newton communicated with alien beings from another planet ?

[@BrenoStrato]

y/x = 1. tg45⁰ = 1.

[@CarlWinter-oy8uf]

@@BrenoStrato DO NOT UNDERSTAND HIROGLYPHCS !--you fail to explain clearly ---calculus ??

[@BrenoStrato]

@@CarlWinter-oy8uf You need to understand other Mathematics concepts - such as arithmetic, algebra, functions, trigonometry, geometry - before understanding Differential and Integral Calculus. Without having a base in these areas, it is difficult to advance. But in response to your question, YES, in the straight line equation y = x, for each value of x you will have a y of the same value. And YES, the angle between the straight line y = x and the x axis is 45⁰. You can guess this intuitively, or you can use trigonometry to deduce the angle. As y/x will always be equal to 1 in this case, the tangent will be 1, which corresponds to an angle of 45⁰.

[@BrenoStrato]

@@CarlWinter-oy8uf Once you know the equation that defines the line, simply integrate x - from 0 to 6. To do this you need to know the rules of integration. In this case, just multiply x • x/2 and you will have x²/2. So the integral of x is x²/2. But you want the area between x = 0 and x = 6, so you replace the x of the integral with the upper and lower limits and you will have A (area) = 6²/2 - 0²/2 = 36/2 - 0 = 18.

[@CarlWinter-oy8uf]

@@BrenoStrato Tangent of 1 is equal to 45 degrees ? ---I only question this logic ---as at school I was told strictly to get lost if you dont understand calculus---- now at 75 --I still have questions ----am I being ridiculous ? ---all I want is simple clarification !

[@hmaines5169]

Not to be that guy but when you are showing the curve y=x^2 and finding the area from zero to 4, the equation for the that line is actually X^2+2 dx.

[@hmaines5169]

which then becomes (x^3)/3 + 2x

[@argonwheatbelly637]

The graph looks more like y=⅓x+2

[@mestaphadawakan7987]

La fonction Y=x2+b car x=0 y=b voir graphe.

[@adgf1x]

integration=18=ar.

[@chuckmccown8893]

What language is the word AERA?

[@FrankD71864]

Aera is a simple error in math.

[@josephlaura7387]

18

[@adrianvasile4683]

It took 17 min to explain something you could do it in 3 minute.

[@mboganene]

This video is not for advanced learners.

[@adrianvasile4683]

@mboganene Explanations for somebody who does not know much should be simple and concise. You lose them after several minutes. There is a different reason here: a long video is more profitable for you.

[@chuckiemeister]

Agree, i've seen the sky's videos, he does wait too much talking then it's necessary and constantly repeats himself over and over.

[@markleeks1116]

6 x 6 = 36 36 / 2 = 18 I thank you

[@emadramadan6510]

Taking too long repeating you could have given 10 examples

[@XZeroOneArmour]

ummm...where is ur advanced calculus tutorials plz ? stuff with trigonometric functions etc.

[@승수노-z3e]

diaghnol?root 72

[@wasimahmad-t6c]

6×6÷2=18

[@mintusaren895]

Quardrinal ase ni , nadi sangam hoi ni. Eta sagar sangame naba kumar er kotha.

[@sureshperumal7464]

Your video is very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very lengthy.

[@sinjhguddu4974]

Teaching technical subjects often fails simply because tutors avoid or skip explaining the whys. For, example what would be so hard about explaining why 1 and not 2 or 3, is added to the power? This would clear alot of fog from the learners mind and prevent them from immediately logging out...

[@martinhicks6020]

AERA? - maybe AREA.

[@DeeGee61]

Thanks for nothing, troll.

[@brettholden3763]

A hopeless explanation when it is apparent that according to the equation y0 should be 0 and y4 should be 16. The graph doesn't represent the equation.

[@modinali3987]

Please don't repeat the same thing again and again.

[@OsvinZacarías-v6q]

So confusing🫠🫠🫠🫠I didn’t understand and I what it 3 times

[@CarlWinter-oy8uf]

Geometry was ok -I excelled ---but calculus ? ---very confusing !

[@CarlWinter-oy8uf]

At 16-:00. --you suddenly stop ---just when we are waiting for the answer --and you start talking about your calculus courses on offer ---?? HOW ABOUT FINISHING THE SUM? X cubed over 3 MINUS. X cubed over 3----Whats the answer ?? huh ?

[@CarlWinter-oy8uf]

We Give up. !