I am 72 years old and enjoy substituting in high school science and math classes. I am in the process of teaching myself calculus. I have an appreciation of limits and derivatives but want to understand how and why integration works. Every video I have watched (and I have watched plenty) showed how to perform integration but none in a manner that made me feel that I really understood the concept. This is the best explanation I have seen thus far. Math books labor over formulas and some videos seem to want you to accept the concept because it is so. The implication being, if you do not understand, it is because you are either not trying hard enough or just not bright enough. This video really want us to understand integration. Thank you. For those still struggling with it, relax and do not stress. Watch it again, in a couple of days. Your subconscious mind will work on it, in between views. If a 72 year old can get it, you can too.
I can't thank you enough for your colossal contribuation to my understanding of maths and physics. I'm a medicine doctor but I'm fascinated by physics and chemistry. I've read a dozen books about physics but the authors focused on the equations and the formulas rather than the phenomena themselves. Your videos are way better. Thank you again.
If I'm not mistaken Prof. Dave takes a more of a Top Down approach giving the over all first. I agree great work P. Dave though I tend to use it as more of a supplement !
Watched a 2 hour lecture on this, did not understand a single thing. Watched 9 minutes of this video and it instantly cleared up the doubts I had. Thank you!
We are the same age more or less. I studied electronics engineering while hated math all my life. Just decided to binge watch all math from start to finish. With a different viewpoint and mostly with this amazing presentation, calm voice, would love it as should. Thanks for the inspiration.
The "dx" actually does have a purpose. It is delta-x from the summation, in the limit as n -> infinity. It is the "width" of a "slice" of the region under the curve, which when multiplied by f(x), the "height", gives us the area of the slice.
well he's saying that since dx is infinitesimally small as n -> infinity, it is a dependent variable and "not important" for the Theorem and inverse relationship of differentiation and integration which he goes on to explain.
It also is an essential part of evaluating an integral, as it tells you in what terms to integrate something in. For instance, if you were to integrate the derivative of y as expressed as dy/dx, you could so so in the following manner. treating derivative as a quotient: ∫ (dy/dx) dx dx denominator and dx at end of integral cancel out ∫ dy (note: this now means that you're integrating in terms of y as an "input variable" or "with respect to y") rewrite integrand as 1y^0 ∫ 1y^0 dy apply power rule = (y^1)/1 + C = y + C This is a more intuitive way to understand how integrals of derivatives return the original function and also understand why you can express a derivative as a quotient in such a way.
dx does have a meaning! It means "a little bit of x" as change in x approaches 0. It is the width of the rectangle. Now, dy/dx = f(x), so dy=f(x)dx. The Sum (integral) of dy, then, is the Sum (integral) of f(x)dx, which equals y + C.
I cannot even begin to express how thankful I am to have you as a resource, it feels good to actually understand something. Thanks for doing what you do, Prof Dave!
If you are a varsity student like me and the textbooks and lectures weren't helping so youtube brought you here, you are safe, just continue watching the series. Thank you Dave.
Can't thank you enough . I've spent entire days trying to self study and search online explanations but you're video explained everything in less than 20mins
If you use the example of velocity (deriv of position) and position, when you want to find the displacement on the velocity graph, you find the area from a to b. When you want to find the displacement on the position graph, you do s(b) - s(a). Therefore, the integral (area) from a to b on the velocity (derivative) graph yields the same result as the s(b) - s(a) (antiderivative graph)
awesome...first time after a zillion attempts i can now fully understand the basic building block of calculus which without your valuable demo i STILL would have been left in the dark...thank you SO much. !! from a very happy math devotee..!!
this is good stuf. my math school teacher illegally didn't teach us everything and now i'm in college trying to learn this for the first time, way after than it should've been but i was only able to understand it now, THANK YOU DAVE
Tell how prof Dave gave me a better understanding of the FTC than my professors that I pay to teach me. Thanks so much, I think I might pass Math now :)
@@ProfessorDaveExplains So that's what the "playlist" feature is all about. I suppose I ought to pay more attention to how this youtube thing works, heh, heh. Thanks.
8:05 The integral of f(t)dt would be g(t) not g(x). x beinf the upper limit doesn't make the anti derivative being dependent of x as if it would be a variable!
Great tutorial! You make calculus easy. But what can finding area under a curve actually tell you. What useful information can I get by finding the area under the curve? I would find it really interesting how can I use this in real life.
Drive a car? Let's take displacement, which is the 0th derivative of position. The 1st derivative is velocity, and the 2nd derivative is acceleration. ( followed by: jerk, snap, crackle and pop..yes, it's true, and no, I'm not working for Kellogs😂😂). You take the integral of acceleration to get the velocity and the integral of velocity to get displacement. in physics this is done with the three Newtonian equations. use the slope of a velocity vs time graph to give you acceleration. If you have acceleration, then you use the area under the acceleration vs time graph to calculate the change in velocity. Pretty cool that your speedometer is giving you the instantaneous velocity( well, minus the vector quantity of direction.)
Thank you for focusing on the “why” in this video. The formulas are great and all, but if you don’t understand why you are using them, they are practically useless.
This is one of those videos where it just snaps into place in your brain, "I understand calculus". Officially feel like neo when he said " I know kung-fu".
I really wish India adopted this kind of an education system...here students who wanna know their topics in depth are called oversmart and you wish i were kidding.
Because the upper limit of integration is not the same as the variable of integration. It is common to use capital and lowercase to distinguish these, but in this example, he chose to use x as the upper limit, and t as the variable of integration.
because it's in the limit of infinity, if you have an infinite number of infinitely thin rectangles, they become the area under the curve. check out the tutorial before this that introduces the concept of integration!
@@ProfessorDaveExplains Ok thanks for the reply, but where is that tutorial? And why is it that the primitive (or anti-derivative) of a function gives the area of that function?
So, what is it about THIS particular inverse relationship that makes it, for many, far less intuitive than those of addition/subtraction and multiplication/division? Is it because people simply don't understand what functions are? Should such concepts be taught earlier than say, high school or college, so the concepts have the requisite time to sink in? Or is it because when all this is typically taught, they just give a set of steps to follow, i.e. dump an algorithm in your lap and tell you to "just do it?" After trying to help my own kids with all this, I"m trying to figure out HOW to help them attain this intuitive understanding of the FTC. Thing is I can't remember what the stumbling blocks were for me. There was no lightbulb flash of insight, all I can think is that it all somehow eventually clicked, but it took a while, and was so gradual that I don't know what that switch was. I'd be interested in hearing others' ideas about this.
It's very abstract. But I think focusing on derivative as rate of change and integration as area under the curve makes them slightly more concrete, though still not obviously inverse.
for me the switch was people don't know how to read math. it isn't like reading a sentence. its like reading an equation is like reading a sentence where every letter in a word stands for a whole paragraph and for me the switch was learning the basics of logs and their inverse functions. and like you said teaching functions better and earlier and relating it to something concrete like mechanics earlier. this video was the cornerstone of my understanding of calculus:
The question is still what does slope has to do with area ? What is the link between medium height of a carve and anti-differential? How segma turns to a function , how. This summation is reduced ? Add to this what is "really" the difference between area and step on x or y axis ? That means can we think of area as an illusion and meaningless magnitude !!! ++ when we talk about area we have to remember that we define area by the mean of square shape. so area is the number of squares. and what is square ? Simply nothing !!!
Because now that we have the theorem established, we don't need each individual student to know how to prove it. You can look up "proof of the fundamental theorem of calculus" if you are curious to know the details.
I am 72 years old and enjoy substituting in high school science and math classes. I am in the process of teaching myself calculus. I have an appreciation of limits and derivatives but want to understand how and why integration works. Every video I have watched (and I have watched plenty) showed how to perform integration but none in a manner that made me feel that I really understood the concept. This is the best explanation I have seen thus far. Math books labor over formulas and some videos seem to want you to accept the concept because it is so. The implication being, if you do not understand, it is because you are either not trying hard enough or just not bright enough. This video really want us to understand integration. Thank you. For those still struggling with it, relax and do not stress. Watch it again, in a couple of days. Your subconscious mind will work on it, in between views. If a 72 year old can get it, you can too.
You are awesome! Thanks for inspiring me!
To truly understand this theorem it takes years of work, hence why it took theorists years of work to derive these formulae.
I am 63.
I tried leraning from guide,at teen ages,because no schools here in my land.
Now after 40 years I am making it from youtube videos.
Such an inspiration sir! Are you still there? I'm currently 17 and my dad is 74 and I wish he were as cool as you (╯︵╰,)
👏
I can't thank you enough for your colossal contribuation to my understanding of maths and physics. I'm a medicine doctor but I'm fascinated by physics and chemistry. I've read a dozen books about physics but the authors focused on the equations and the formulas rather than the phenomena themselves. Your videos are way better. Thank you again.
If I'm not mistaken Prof. Dave takes a more of a Top Down approach giving the over all first. I agree great work P. Dave though I tend to use it as more of a supplement !
Accepting an idea that I don't actually understand is stress.
You're saving and transforming lives.
God bless you Prof🙏
Watched a 2 hour lecture on this, did not understand a single thing. Watched 9 minutes of this video and it instantly cleared up the doubts I had. Thank you!
We are the same age more or less. I studied electronics engineering while hated math all my life. Just decided to binge watch all math from start to finish. With a different viewpoint and mostly with this amazing presentation, calm voice, would love it as should. Thanks for the inspiration.
The "dx" actually does have a purpose. It is delta-x from the summation, in the limit as n -> infinity. It is the "width" of a "slice" of the region under the curve, which when multiplied by f(x), the "height", gives us the area of the slice.
uncool @John Doe
You mentioned "when multiplied by the f(x)", but Dave only stated that it's useless on its own.
yes
well he's saying that since dx is infinitesimally small as n -> infinity, it is a dependent variable and "not important" for the Theorem and inverse relationship of differentiation and integration which he goes on to explain.
It also is an essential part of evaluating an integral, as it tells you in what terms to integrate something in. For instance, if you were to integrate the derivative of y as expressed as dy/dx, you could so so in the following manner.
treating derivative as a quotient:
∫ (dy/dx) dx
dx denominator and dx at end of integral cancel out
∫ dy
(note: this now means that you're integrating in terms of y as an "input variable" or "with respect to y")
rewrite integrand as 1y^0
∫ 1y^0 dy
apply power rule
= (y^1)/1 + C
= y + C
This is a more intuitive way to understand how integrals of derivatives return the original function and also understand why you can express a derivative as a quotient in such a way.
dx does have a meaning! It means "a little bit of x" as change in x approaches 0. It is the width of the rectangle. Now, dy/dx = f(x), so dy=f(x)dx. The Sum (integral) of dy, then, is the Sum (integral) of f(x)dx, which equals y + C.
wow, beautifully explained. thank you!
yes, OMG so much clarity in so few words
I cannot even begin to express how thankful I am to have you as a resource, it feels good to actually understand something. Thanks for doing what you do, Prof Dave!
So many people told me calculus was hell and hard. But since this is extremly easy for me, it must be the great teacher dave
If you are a varsity student like me and the textbooks and lectures weren't helping so youtube brought you here, you are safe, just continue watching the series. Thank you Dave.
Can't thank you enough . I've spent entire days trying to self study and search online explanations but you're video explained everything in less than 20mins
Yes but why does the antidervitive gives you the area under the curve
If you use the example of velocity (deriv of position) and position, when you want to find the displacement on the velocity graph, you find the area from a to b. When you want to find the displacement on the position graph, you do s(b) - s(a). Therefore, the integral (area) from a to b on the velocity (derivative) graph yields the same result as the s(b) - s(a) (antiderivative graph)
put it this way = { change in f(x)/change in x} multiplied by x, is just change in f(x) which is the area under the curve.
does that help?
@Manny K Thanks so much for posting this
Thanks!
Will need this for next semester when I take Calc! Good stuff as always
Thank you sir for your dedication and for making this free! 🙏
So clear and simple
I love it
thank you professor Dave
awesome...first time after a zillion attempts i can now fully understand the basic building block of calculus which without your valuable demo i STILL would have been left in the dark...thank you SO much. !! from a very happy math devotee..!!
No, you don't! You don't understand at all because Prof. Dave does not understand!
ua-cam.com/video/1URPfy4cbDA/v-deo.html
this is good stuf. my math school teacher illegally didn't teach us everything and now i'm in college trying to learn this for the first time, way after than it should've been
but i was only able to understand it now, THANK YOU DAVE
You knocked that one out of the park professor D!
You all that bro", keep jammin'.
Tell how prof Dave gave me a better understanding of the FTC than my professors that I pay to teach me. Thanks so much, I think I might pass Math now :)
Still don't get it....😓😭 I wish I was smart matematically.
I will continue watching, maybe one day I will have my Eureka moment!
Thanks Prof.Dave.
yes sometimes you just have to watch more than once, you'll get it!
Ania Wo you’ll definitely get it!! Don’t give up, you may be one more video or one problem away from it clicking and making sense!
Oh yeah. I'm the same way. Sometimes I need to review a video several times before it clicks. No shame in it
@@EvilSandwich Aww, thank you guys ❤
man.. this vid made me relax and chill
I like your way of teaching. Where is the best place to start with your videos for a thorough understanding of calculus?
Just pull up the calculus playlist and watch from start to finish!
@@ProfessorDaveExplains So that's what the "playlist" feature is all about. I suppose I ought to pay more attention to how this youtube thing works, heh, heh. Thanks.
this is gold man, thank you
2:00 dx actually has a meaning , fx gives the height of rectange and dx is the width (thats how one of my teacher explained it ) . Is it correct?
great explanation, way clearer than khan academy
Excellent explanation for the fundamental theorem of calculus!
very clear and straight to the point, thank you !
Great "ladder" analogy! Thanks!
I had to calculate the area under the curve using limit of riemann sums :( for my final, if only I could use ftc part 2
8:05 The integral of f(t)dt would be g(t) not g(x). x beinf the upper limit doesn't make the anti derivative being dependent of x as if it would be a variable!
I watched it.
I call it the derivative ladder. Once you think of a list of functions that way, you're all set.
great video sir..
This is gold, thank you! :)
blud out here saving livesssss
You are the best Dave ! Thank you so much ! You are really amazing your methods are incredibly good !
Nice video, and well editings!
Great tutorial! You make calculus easy. But what can finding area under a curve actually tell you. What useful information can I get by finding the area under the curve? I would find it really interesting how can I use this in real life.
Drive a car? Let's take displacement, which is the 0th derivative of position. The 1st derivative is velocity, and the 2nd derivative is acceleration. ( followed by: jerk, snap, crackle and pop..yes, it's true, and no, I'm not working for Kellogs😂😂). You take the integral of acceleration to get the velocity and the integral of velocity to get displacement. in physics this is done with the three Newtonian equations.
use the slope of a velocity vs time graph to give you acceleration. If you have acceleration, then you use the area under the acceleration vs time graph to calculate the change in velocity. Pretty cool that your speedometer is giving you the instantaneous velocity( well, minus the vector quantity of direction.)
good question! never stop asking why!!!
You're a lifesaver!!!
Thanks
Thank you! Very well explained!
thank you so much
6:00 the addition and multiplying ladder is fine..however the integration ladder makes little sense...
your videos were very helpful and lucid for me .thanks for making such awesome videos
Dx is the length of each rectangle
This is actually interesting
Giga chad math W
The video is very interesting and nice
Plus a constant!!!!!
Done.
Thank you for focusing on the “why” in this video. The formulas are great and all, but if you don’t understand why you are using them, they are practically useless.
How performing integration adds area of those tiny rectangles means integration just change slope into function
Hi i didn't understand how area under curve=anti-derivative. Any other videos on that ?
maybe?: ua-cam.com/video/Pkwk3ZP2xOg/v-deo.html
It is incredible that such a concept didn't exist for millenairies until Newton and Leibniz discovered it.
This is one of those videos where it just snaps into place in your brain, "I understand calculus". Officially feel like neo when he said " I know kung-fu".
8:08
why is g'(x)=f(x)?
g(x) is an integral, an antiderivative. Taking the derivative of an antiderivative gives you the original function.
I really wish India adopted this kind of an education system...here students who wanna know their topics in depth are called oversmart and you wish i were kidding.
From 7:05 why are we working with t
Because the upper limit of integration is not the same as the variable of integration. It is common to use capital and lowercase to distinguish these, but in this example, he chose to use x as the upper limit, and t as the variable of integration.
A ab are area
just one word, KING :D
1:01 is exactly where it puzzles me.
How do you justify the transformation from the discrete sum Σf(x)Δx to the continuous sum ∫f(x)dx ?
because it's in the limit of infinity, if you have an infinite number of infinitely thin rectangles, they become the area under the curve. check out the tutorial before this that introduces the concept of integration!
@@ProfessorDaveExplains
Ok thanks for the reply, but where is that tutorial?
And why is it that the primitive (or anti-derivative) of a function gives the area of that function?
it's in the mathematics playlist, and also in the shorter calculus playlist, it's all in there
Dx=delta x
And the limit and submission sign are replaced by an integration sign f(x) remains same
what is the equivalence of FTC
It is not fundamental it is force
Thank you very much Professor Dave, you are a super Teacher.
So, what is it about THIS particular inverse relationship that makes it, for many, far less intuitive than those of addition/subtraction and multiplication/division? Is it because people simply don't understand what functions are? Should such concepts be taught earlier than say, high school or college, so the concepts have the requisite time to sink in? Or is it because when all this is typically taught, they just give a set of steps to follow, i.e. dump an algorithm in your lap and tell you to "just do it?" After trying to help my own kids with all this, I"m trying to figure out HOW to help them attain this intuitive understanding of the FTC. Thing is I can't remember what the stumbling blocks were for me. There was no lightbulb flash of insight, all I can think is that it all somehow eventually clicked, but it took a while, and was so gradual that I don't know what that switch was. I'd be interested in hearing others' ideas about this.
It's very abstract. But I think focusing on derivative as rate of change and integration as area under the curve makes them slightly more concrete, though still not obviously inverse.
@@ProfessorDaveExplains Yeah, tried that already, didn't help.
for me the switch was people don't know how to read math. it isn't like reading a sentence. its like reading an equation is like reading a sentence where every letter in a word stands for a whole paragraph and for me the switch was learning the basics of logs and their inverse functions. and like you said teaching functions better and earlier and relating it to something concrete like mechanics earlier. this video was the cornerstone of my understanding of calculus:
ua-cam.com/video/mpkTHyfr0pM/v-deo.html there are like 7 of thease in a row
make it a war game and teach your children chess.
great
Thank you so much sir.may God bless you.you are saving so many lives & grades 😘😘. you are looking like a actor Chris evens 🤗🤗
why does no one explain or show what F(b) - F(a) really is doing in terms of the antiderivative.
Uuuuuuh isn’t the bracket notation the same thing as Af(x) or f(x1) - f(x2)? (A is the delta symbol)
The question is still what does slope has to do with area ?
What is the link between medium height of a carve and anti-differential?
How segma turns to a function , how. This summation is reduced ?
Add to this what is "really" the difference between area and step on x or y axis ? That means can we think of area as an illusion and meaningless magnitude !!!
++ when we talk about area we have to remember that we define area by the mean of square shape.
so area is the number of squares. and what is square ? Simply nothing !!!
sir make a video about equilibrium condition in Civil engineering plzz sir plz
You mean forces add up to zero and torques add up to zero? Or is there a different meaning of equilibrium?
Why wasn't there any proof here. You seemed to just state the two are equal.
Because now that we have the theorem established, we don't need each individual student to know how to prove it. You can look up "proof of the fundamental theorem of calculus" if you are curious to know the details.
Undoes
I love you
Finally :D
Still hate Newton lol
The idea comes from india..derivatives and integrale were conceived by ARYABATTHA BRAHMAGUPTA BASHKARA SECONDO MANJULA MADHAVA etc..
Mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
You are the best Dave ! Thank you so much ! You are really amazing your methods are incredibly good !