This is actually the best video I’ve found so far that explains it so simply and elegantly. Good job! Will definitely recommend to anyone struggling with this proof.
WOW!! I didn’t know that way of seeing that the area under a curve is just its antiderivative. I will save this video :) you saved me in an assignment and now YT is recommending yours videos (I am glad it does 👍🏻👍🏻)
Yeah both derivatives and integrals are basically just geometric interpretations of slope. That is what the "little shapes" mentioned in the video actually are. Geometric interpretations taken to be thinner and thinner until it becomes the integral instead...
@@seagulyus9251 The integral is definitely not a geometric interpretation of the slope. An integral is the continuous analog of asummation. Its geometrical interpretation would be the area under a curve which is not very precise since this only the case when dealing with R to R functions. There are also many types of integrals like line integral or complex ones where the result might be complex which wouldn’t make any sense if we think about it in therms of area.
@@yosoy9766 I was talking about real functions, specifically the original formulas used to build derivative and integration were visualized using geometric interpretations of slope and trapezoids. It is just that how I was taught integrals was...going from geometric interpretations to cutting the distance between x's down to delta x and then we were there. I highly doubt anyone that anyone seeing this video as helpful would deal with either line integrals or complex integrals enough that explaining the detailed nuances of each would be worthwhile y'know? Especially when real integrals are the ones students deal with, and geometric is basically the way to help most of them see what it is they need to do.
I can now imagine integration and differentiation in my mind 🙏😭😭 you can understand the pain of not understanding calculus by teachers who just don't know how to explain things and youtube where videos are in another language ( I know enlish but of course , mother tongue is mother tongue ) Thank you so so much ❤️
This is the best explanation of connection b/w integration and area under a curve. God bless u sir. Wish u were my math teacher in college. I would have loved calculus instead of hating it
This is amazing, bro. Thank you soooo much. Your simple but elegant explanation makes me understand the relationship between integration and area under graph (In school, our teacher only taught that's how integration works without any explanation XD)
It was literally awesome man. I was insanely looking for the answer to the question "why is integral of a function the area under it's curve?'' Finally got the answer..... Thanks a lot brother...
You've given me a breakthrough. I have "understood" the integral for over 3 years now, but never fully reconciled the graphical correlation between the antiderivative transform of the base function and the area under that function's curve. Thank you so much.
Thank you so much! I'm self-studying calculus and I was scouring the internet for an explnation as to how an indefinite integral actually relates to area, and every source kept repeating "antiderivative." They did not relate the term "antiderivative" to area at all, and this is the first video which actually bridged the gap for me. Thank you so much for the video! Just to clarify, an indefinite integral of a function will provide you the area of the original function from x=0 to any x-input you specify directly into the integral function, right?
Well an indefinite has the constant C, so its not like it really gives a area, but when it has two bounds/limits the constant C isn’t an unkown value anymore because you can compare both of the anti derivatives (two bounds) But I cant so more cuz I also aint a pro at this yet
A(x+h)-A(x) doesn't have to be between f(x)*h and f(x+h)*h but could instead be bigger or smaller than both right? Also if the graph goes downward surely the greater than signs would have to be reversed so with a not specific graph like the one you used for the example you wouldn't be able to make an assumption like that.
Excellent video! What I didn't understand is the last part. Why integrating A'(x) gives A(x)? Is there a simple explanation for this, knowing just the definitions of a limit and a derivative?
@@thecalamity278 Yes, I know that already because I was told to, but is there a simple explanation for why integration and differentiation are reverse operations?
@@nickxyzt sorry I don't think I quite get your question? In my understanding one way to define an integral is just the antiderivative as it's what you do to get back to a primitive function after differentiation. If you're asking why area and gradient come from inverse functions, that's a very good question and I don't know the answer!
@@thecalamity278 Yes, that's my question 😀 The definition for the integral is the area under the curve (as defined by Leibnitz), and the definition for the derivative is its slope (as defined by Newton). However, they are inverse functions, but I don't know why, and I couldn't find an explanation!
since we defined A(x+h)=h*f(x+h). the definition of area of a rectangle is l*w so by saying A(x+h) by this very operation it makes it the area function (probably would sound better if I explained verbally)
"Why does integrating a function give area under its curve?" Because of the MEAN VALUE THEOREM and none of the nonsense in your video! ruclips.net/video/nKv3IMhKlxk/видео.html
Dude, I've have yet to see an explanation as simple, concise and well done. So clear! Thanks!
This is actually the best video I’ve found so far that explains it so simply and elegantly.
Good job! Will definitely recommend to anyone struggling with this proof.
Gotta love the squeeze theorem. Great video.
Sandwich theorem is more likeable lol
I don't think I can express how much I am thankful for this 💫💫
WOW!! I didn’t know that way of seeing that the area under a curve is just its antiderivative. I will save this video :) you saved me in an assignment and now YT is recommending yours videos (I am glad it does 👍🏻👍🏻)
Yeah both derivatives and integrals are basically just geometric interpretations of slope. That is what the "little shapes" mentioned in the video actually are. Geometric interpretations taken to be thinner and thinner until it becomes the integral instead...
@@seagulyus9251 The integral is definitely not a geometric interpretation of the slope. An integral is the continuous analog of asummation. Its geometrical interpretation would be the area under a curve which is not very precise since this only the case when dealing with R to R functions. There are also many types of integrals like line integral or complex ones where the result might be complex which wouldn’t make any sense if we think about it in therms of area.
@@yosoy9766
I was talking about real functions, specifically the original formulas used to build derivative and integration were visualized using geometric interpretations of slope and trapezoids.
It is just that how I was taught integrals was...going from geometric interpretations to cutting the distance between x's down to delta x and then we were there.
I highly doubt anyone that anyone seeing this video as helpful would deal with either line integrals or complex integrals enough that explaining the detailed nuances of each would be worthwhile y'know? Especially when real integrals are the ones students deal with, and geometric is basically the way to help most of them see what it is they need to do.
amazing man, never give up teaching please, you're doing great work!
Keep it up man. I can't help but remember the pilots whenever I hear your voice
I can now imagine integration and differentiation in my mind 🙏😭😭 you can understand the pain of not understanding calculus by teachers who just don't know how to explain things and youtube where videos are in another language ( I know enlish but of course , mother tongue is mother tongue ) Thank you so so much ❤️
This is amazing! Thanks for the enlightenment.
This is the best explanation of connection b/w integration and area under a curve. God bless u sir. Wish u were my math teacher in college. I would have loved calculus instead of hating it
I understood the rectangle method but this method using the squeeze theorem just reinforces everything and makes it much more clear. Thanks!
keep doing this man, you're great!
This is amazing, bro. Thank you soooo much. Your simple but elegant explanation makes me understand the relationship between integration and area under graph (In school, our teacher only taught that's how integration works without any explanation XD)
Glad it was helpful!
Beautiful explanation
A very eye opening video. Keep it up friend. God bless you❤
Neat, precise and profound. Thank you so much ❤️❤️❤️
Glad it was helpful!
It was literally awesome man. I was insanely looking for the answer to the question "why is integral of a function the area under it's curve?'' Finally got the answer.....
Thanks a lot brother...
Thanks! Now it all makes sense why this works! I love this way of thinking of it and this video is well worth watching for a better understanding.
Thank you for your worthy explanation!
Thank you for this Video it’s the best explonation that I have seen so far
Very nice. ❤
Thank you this is exactly the type of answer I was looking for
This is awesome!!!
Thank you so much, very great video
Thank you for the video :)
that was beautiful, thank you for sharing
Glad you enjoyed it
This makes so much sense!
holy shit you blew my mind im starring this video in case i forget
Thanks for sharing
Niceee !! Love it
bro this is amazing
I underestimated the squeeze Theorem in my high school, now I'm here
Thanks it helps
Thank you!
Hella nice man
you're awesome
great video! I have a question tho, why can u set f(x)h less than or equal to A(x+h)-A(x) instead of it just being f(x)h < A(x+h)-A(x)?
It will be equal if f(x) is a horizontal line
You've given me a breakthrough. I have "understood" the integral for over 3 years now, but never fully reconciled the graphical correlation between the antiderivative transform of the base function and the area under that function's curve. Thank you so much.
Thank you so much! I'm self-studying calculus and I was scouring the internet for an explnation as to how an indefinite integral actually relates to area, and every source kept repeating "antiderivative." They did not relate the term "antiderivative" to area at all, and this is the first video which actually bridged the gap for me. Thank you so much for the video! Just to clarify, an indefinite integral of a function will provide you the area of the original function from x=0 to any x-input you specify directly into the integral function, right?
Well an indefinite has the constant C, so its not like it really gives a area, but when it has two bounds/limits the constant C isn’t an unkown value anymore because you can compare both of the anti derivatives (two bounds)
But I cant so more cuz I also aint a pro at this yet
what a beast man😂💯
Tysmmmmmm
A(x+h)-A(x) doesn't have to be between f(x)*h and f(x+h)*h but could instead be bigger or smaller than both right? Also if the graph goes downward surely the greater than signs would have to be reversed so with a not specific graph like the one you used for the example you wouldn't be able to make an assumption like that.
Only yt vid ive liked
Woahh, I did not know math can do such thing😮
Excellent video! What I didn't understand is the last part. Why integrating A'(x) gives A(x)? Is there a simple explanation for this, knowing just the definitions of a limit and a derivative?
Integration is just the opposite of differentiation, so integrating a derivative gives the original function
@@thecalamity278 Yes, I know that already because I was told to, but is there a simple explanation for why integration and differentiation are reverse operations?
@@nickxyzt sorry I don't think I quite get your question? In my understanding one way to define an integral is just the antiderivative as it's what you do to get back to a primitive function after differentiation.
If you're asking why area and gradient come from inverse functions, that's a very good question and I don't know the answer!
@@thecalamity278 Yes, that's my question 😀 The definition for the integral is the area under the curve (as defined by Leibnitz), and the definition for the derivative is its slope (as defined by Newton). However, they are inverse functions, but I don't know why, and I couldn't find an explanation!
I have a question. This one in the half bounded between f (x) and f(x+h). Why is it A'(x) not A'(x+h)
Because as h tends towards 0, the A(x + h) just becomes A(x) I think
you have to understand the limit definition of a derivative first
epic
I don't know why in math learning videos, you start concentrating extremely when you don't know where something rised from
How would you prove that a function defining the area for a given function even exist??
since we defined A(x+h)=h*f(x+h). the definition of area of a rectangle is l*w so by saying A(x+h) by this very operation it makes it the area function (probably would sound better if I explained verbally)
@@MathsPhysicshelp Yeah that indeed makes sense. Thanks Man!
Why integrating a function gives the area under its graph? Because that's why integration was defined.
"Why does integrating a function give area under its curve?"
Because of the MEAN VALUE THEOREM and none of the nonsense in your video!
ruclips.net/video/nKv3IMhKlxk/видео.html
I've watched your video and I can't see how this working is incorrect, Could you Explain what's wrong?
Bro I think he is idiot, he know only abuse on others.
Defining calculus in many ways is obviously possible because truth can be find by many ways .
Would have walked in and straight off the back guess pi, pi/2, and pi/4 if those didn’t work definitely google lens that mf