If you want clear cut explanation , its here ukposts.info/have/v-deo/j6OChax_pGqdtKs.html

correction ukposts.info/have/v-deo/j6OChax_pGqdtKs.html

No you didn't. In order to solve this, you need to know the trick .You can't intuitively come up with the idea that you can work with 3D objects in a 2D cartesian plane. Never .This formula wasn't derived originally from this method. So it's just a trick .People know that it works but probably don't know why it works.Or at least you had to have the knowledge of 'disc' method.

​@@user-wd7xu4mf8oMaybe he or she is just smarter than you and they actually did do it.

@@user-wd7xu4mf8o ..i beg to differ... im absolutely sure this is how he/she solved it... its a methodd that works for every rotational volume... no matter how complex it is... in some cases its the only way to solve it mathematically.. ...this was also something i myself came up with long before rotational volumes was even thought.. so yes im absolute certain he /she figured it out... ..its also what sets some student apart.. there is two types of students, one who know why, and one who always get the right asnswer, its the once who know why that change the wourld (to quote myself) in a nutshell V = πy^2dx; y=lim(F'(x) * F(x) )

@@Patrik6920 so you didn’t have the idea of "Rotational volume " right?Wow you're smart.You're so different from other students. I guess you could do Algebra without learning how to add, subtract or multipy.

@@siamsama2581 wooh!what a smart guy he is!He could do cartesian geometry without having the knowledge of slopes. People like you don’t progress in life.Move on and sit with pen and paper and try to figure out if you're also smart.

Hu Tao?

@@capt.price1419 yes, it's me Hu Tao, Funeral Business was not going well so Mr.Zhongli sent me to high school, It's so boring here, here no fire butterflies 🦋 🔥

Because some of the water had evaporated by the experiment’s end…

I was 🤯 when I figured out, years ago, that a cone was a pyramid with a circular base.

Your formula makes no assumptions about the shape of the base, so you may as well generalize this further: Given you have a two-dimensional figure where the area is known. Now imagine you create a three-dimensional figure by carrying the edges of the 2-D figure to a point P not in the plane of the 2-D figure. For example, if the 2-D figure is a circle, you'll get a cone; if the 2-D figure is a hexagon, you'll get a hexagonal pyramid, etc. Then the area of the 3-D figure is going to equal Bh/3, where B is the area of the base and h is the perpendicular distance from the plane containing the 2-D figure to the point P. And now, what the heck, why stop at three dimensions? Given a 3-D object with a known volume which is turned into a 4-D figure by connecting it with a point P in the fourth dimension, then the 4D-volume of the 4-D figure is Vh/4, where V is the volume of the 3-D figure and h is the perpendicular distance from the space containing the 3-D figure to the point P in the fourth dimension. (I know we can't visualize this because we're stuck in three dimensions, but the math is valid, so the formula must be correct!) And, we can even generalize this for any n-dimensional figure turned into an (n+1)-dimensional figure by connecting it with a point P in the next dimension. The n-Dimensional volume will be Vb/n.

Yeah the cross-sectional method is really useful.....

👍👍👍👍👍❤

I told you Calculus was sweet.

selling icecream IS INDEED better

You should have learned it in calculus II

@@Eric-xh9ee The spanish education system is just built differently... It's bad, I'm not gonna lie.

@@Dantido Huh weird. This is a pretty core concept so it's good to know.

@@Eric-xh9ee Yeah, it's a real shame. We only went over limits, differentiation, undefined integrals, function areas with defined integrals, and some theorems. I guess it's to leave space for matrixes and geometry, which is important, sure, but it's much less fun and more hard work-oriented than analysis in my opinion. I actually ended up looking up more stuff about calculus myself. It's so damn interesting and fun, definitely the best part of math for me.

Always blaming the teacher even if you are stupid

Why don't you blame your parents Go back from where you came 😂😂😂

Honestly I don't think you need integration to prove this volume formula. I'm sure they'd figured it out many hundreds of years ago already and calculus is a kind of modern overkill here

calculus' foundation originated over 5000 years ago in the moscow papyruts or something. I forgot the name but I do remember they were moscowian @@jellymath

Assumes the proof. But true.

A n-dimensional cone will be divided by n

y=r, x=h , this implies that y/x = r/h , from this you can derive y = (r/h)x

how?

its Function when you come h from x axix the equ=r.h/h=r its our y axis

When you rotate the triangle to sweep a circle, the outer edges of the triangle trace a larger circle than the inner, and thus account for a greater portion of the volume. Since the triangle is thinner at the outer edges, the smaller parts of the triangle contribute more to the swept volume. Meaning the volume is less than the 1/2 method you propose. If you do the integral, you realize the correct factor is 1/3.

You must integrate multiplying gives you the volume of a wedge instead

Still doesnt answer why it equals that. Only proves WHAT it equals. Completely different question.

Yes, a cone is a circle whose radius gradually reduces to zero. You've restated the question instead of answering it.

Do you know why the volume of a pyramid is height*width*length/3?

Well you've come to the right place. Watch the video and find out.

The line has a slope of r/h (rise/run) and a y-intercept of 0. So writing it as y=mx+c we get y=(r/h)*x + 0 -> y=(r/h)x

it's "the curve" . it also has the +b part equal to 0, y=mx+b

@@jamescollier3 yeah ik i just forgot that the derivative is just a ratio rise over run and i was thinking why shouldnt m be like a normal number a for example

No, it can only be done with intrigation

It is literally the same, even in terms of clarity it makes absolutely no difference.

@@Ninja20704 But why do people choose to use ⅓πr²h instead of πr²h/3 when dividing by 3 makes more sense for most people?

Troll question!

@@channelbuattv I mean, we do the exact same with like the area of triangle (1/2)*b*h and also volume of sphere (4/3)*pi*r^3. I would believe it is because writing the fraction shows the relationship between the cone and cylinder that the cone is 1/3 of the cylinder (same as with a triangle and rectangle)

so we use the space efficiently. putting everything over small number would leave unusable space

yes , you can find without integration ... euclides book 12 proposition 10 ... based on props before ... anyway in order to understand it is a bit a reasoning alike integrals .... actualy it is based on the fact that you can split up piramide in two smaller piramides of the same shape and two prisms ... , that smaller piramides you can do the same tric .... finaly you get a sum of piramides and prisms ... the greeks say that you can split it up as far as you want and that the sum of the small priamides becomes neglectable ... the other sum is kind of infinite sum in the sence of 1/4 + (1/4)^2+ ( 1/4) ^3 ... which leads to 1/3 ... , anyway the greeks say it a bit different , just to avoid the infinite sum like this ...

a more intuitive approach is to construct a piramide of two piramides of the same shape and two prisms ... if you assume that the two little piramides have 1/8 of the volume of the big one ( since every distance is devided in two ) and the two litlle prisms have the 1/8 of the volume of the surounding prism of the big piramide then you can avoid infinite sum by putting it in an equation where snales bytes its tail .... like 1/8 P + 1/8 P + 1/8 Pr + 1/8 Pr = P and this leads to Pr/3 = P

the approach of euclides book 10 is more or less the following ...o first they proof that the volume of piramide is proportional to its base , and proportional to its height ( not explicitely but indirectely as prisms are also proportional to their base ) .... then they cut 3 piramides out of a prism and proof that these piramides have the same volume ... the most difficult proof where infinity is a bit avoided is the proof where it is stated that the volume of a piramide is proportional to its base ... once proved for piramides , the step to cones is a bit already a sum of very small piramides which in a way seems a bit calculus ...

@@josleurs4345 that's pretty genius, thanks for your reply it was very helpful

Take a cylinder same as cone dimension fill the cone with water, amount of water to fill the cylinder is 3 times of cone volume. Volume of cylinder is pi x r^2 x h

Stop with your cursing. It is ignorant and needless.

cant you see?? like, its literally on the whiteboard

@@NaradaFox he made a mistake and corrected it later. I paused it before he made the correction trying to figure out what i wasn't getting, but in the end it just turned out that he made a mistake

@@robertveith6383 Bro it's the internet who cares if i curse

​@@NaradaFoxDamn ignorant comment. Not everyone is perfect as you in maths.

But why?