Why do integrals always have a dx?

There's so much confusion around dx, especially among Calc 1 and Calc 2 students. And with good reason. dx doesn't start to feel relevant until later on, when you start working with multivariable functions, multiple integrals, and differential equations.
But that's no reason not to at least address dx in an introductory way. In this video, we'll talk about a few of the simplest explanations for what dx does, what it tells us to do, and what it represents.
When you're just starting out with integrals, for the most part, it's okay to think about dx as just notation. It just comes with the integral symbol, and you don't need to know exactly why it's there in order to know that you're supposed to integrate.
Once you start changing variables, it's important to know that the dx always tells you which variable to integrate. That's why you'll see dx with a function in terms of x, dt with a function in terms of t, and dy with a function in terms of y. You'll also need to understand that your limits of integration (if you're dealing with a definite integral), will need to match the variable in your function and the variable in your dx/dt/dy/etc.
When you get to multivariable calculus and multiple integrals, you'll need to realize that seeing dx dy at the end of the integral tells you to integrate first with respect to x, and then with respect to y.
But most importantly, you'll want to understand that dx represents the differential, or the difference between two values of x. It's the distance between two values of x. So, when you use a Riemann sum, or trapezoidal rule to approximate the area under a curve, delta x is the width of each rectangle (or trapezoid).
When you switch from these approximation methods, and start using the integral instead to find exact area, that delta x has to change to dx in order to match the integral. Essentially what you're doing is saying that delta x represents a larger width, which is why you can only get an approximation. But dx represents and infinitely small distance, which is why it gets paired with the integral, and allows you to find exact area, instead of just an approximation.
And that's why you always need a dx whenever you're using an integral.
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[@tomryan9827]

It's amazing how few math teachers actually explain things in English. 30 seconds of talking like a human can clear up years of confusion

[@thedoublehelix5661]

In higher level math classes like real analysis people usually do talk like humans

[@acoleman8783]

Definitely

[@stuart5960]

Hear! Hear!

[@GhhyBbbb]

@@thedoublehelix5661 Hej

[@4bschaum]

@@thedoublehelix5661 i think its the other way around. in higher level math classes you have people that are comfortable with the lingo. my math teacher in 7th grade was egregious with that shit. why would you talk to a bunch on 13 year old little farts like you would talk to master students?

[@KillianDefaoite]

This is something that's always sort of skipped over in introductory calculus classes. Glad to see this video.

[@guitarttimman]

That's only true about community colleges, and that's why I attended Purdue University to learn mathematics. LOL

[@o1dragone]

it's a DLC

[@d9e240]

Maybe I should have gone to Purdue, cause it was skipped over for sure in my math classes. I'm in Calc. II and was totally confused about the notation.

[@erikrodriguez8383]

Yooo I was so confused about dx because it was skipped over in my calc 2 college class

[@jwhine]

@@guitarttimman fuck purdue hoosier daddy

[@uiticus]

The confusion with dx for me was that you can interpret dx as being the derivative of x instead of which variable to integrate. for example in the case dx/dy which interprets to be the derivative of x with respect to the variable y.

[@sheshan8]

Your sound is so soothing.

[@letsherptothederp]

It's fucking annoying, why is she whispering

[@qukoongames5830]

Sheshan Patel voice*

[@okoyoso]

Smooth change requires a smooth voice to explain it.

[@BassByTheBay]

@@letsherptothederp Whispering is speaking without engaging your vocal cords (just using air), so she's not whispering.

[@Grassmpl]

@@okoyoso yeah at least to one order derivative

[@Tamoor622498]

Your voice is so soothing, you need to make a math bedtime story series.

[@xr280xr]

I made it through Calc III in college and never got an explanation for this until now. And it's so simple!

[@ChekMaFresh]

You explained this in a way that was easy to follow and pleasant to listen to. Thank you so much! New sub!

[@kristakingmath]

Thanks for subbing, Chek! :D

[@SuperMaDBrothers]

XD dx*

[@sirjain4408]

hahahahahahahahahahahahhahahahah

[@eylulvv]

@@sirjain4408 omg profile pic bestieeees

[@sirjain4408]

@@eylulvv oh yeah lol

[@Tomas-kq1im]

Lol dx

[@Yazan_Majdalawi]

@@Tomas-kq1im lol

[@BassByTheBay]

I squeaked through a year of calc by relying on short-term memory of the formulas that were fed to us way too quickly on the college quarter system. I would've actually understood it if your content existed back then. 😊

[@davidlourensz4822]

Krista, just so you know and I think you do know, Mathematics is not so hard, it has been hard for me and now I am 56 years old, just the people who teach it don't understand it in the first place because if they did, they could find easy ways to communicate it. I think you do a pretty good job, I will look at what you got in your videos. I feel I really want to master this now and know I got it finally. Thanks for your videos.

[@dashoelorenz2677]

dx simply just means "a little bit of x" ∫ looks like the letter S and it means "sum" or "sum of all" now combine both and you get ∫ dx which means ”the sum of all bits of x”

[@itachi6336]

How do u make an integral sign

[@guitarttimman]

ha ha that sounds hillbilly.

[@guitarttimman]

I hope you don't say UP AIR instead of down the street. :-)

[@SuperYtc1]

Nope, it means as the change in x approaches 0. You can never get to 0, you can only approach it. It’s an idea, a concept, that you approach. Just like if you make a shape with edges and increase the edges until you approach infinity you get closer and closer to a circle. But it’s impossible to have an infinite amount of edges, hence why there is no such thing as a perfect circle. A circle is a theoretical concept you can approach.

[@kylenason]

@@SuperYtc1 wouldn’t any curve have an infinite amount of edges? And if that is the case then by your logic or what you said in the statement above, then wouldn’t that mean there is no such thing as a curve.

[@exor6100]

I was looking for the right way to explain this concept to students and this is the more thorough yet intuitive way I could find. I feel like even my understanding was improved by watching this video. Thank you.

[@kristakingmath]

You're very welcome! So glad it helped! 🤗

[@julioezequiel8935]

Thank you so much, that was one of my biggest doubts. "dx" is one of that stuff that we just pass trought cause its not gonna make we stop the operation. You made my day, thank you !

[@JossinJax]

Ohhh this is great. It's easy to find "how to"s vids, but it's these types that help integrate our understanding. Thanks again!

[@MrChocowocko]

I LOVE THESE VIDEOS. I USED IT FOR MY APPLIED CALC CLASS LAST YEAR AND NOW IT'S HELPING ME WITH MY ANALYTICAL ECONOMICS CLASS THIS SEMESTER. THANK YOU

[@kristakingmath]

+MrChocowocko You're welcome, I'm so glad the videos are helping!

[@amogelangsekhu3906]

At my level of study, option 2 makes the most sense for me. thank you so much, simply the best !!❤️❤️

[@ArthurMorganFTW_RDR]

After watching this video I was absolutely speechless, I mean no one has ever given me such a great explanation. I kinda help my friends, cousins, younger brothers and sisters in their math lessons, if they will ask me about dx I will directly send them to this video..... Thank u once again It was really helpful.

[@kristakingmath]

You're welcome, I'm so glad it helped! :D

[@MakingWavesRF]

I have wondered this for 10 years. This was a great explanation.

[@kristakingmath]

Thank you, Daniel! I'm so glad it helped! :)

[@cicciobombo7496]

Just discovered this channel and i love it, it does explain WHY and HOW something works, not just giving a definition(even if the clearest and easiest definition ever)

[@kristakingmath]

:D

[@joefuentes2977]

The best answer lies with understanding differential forms. In calc 1 and 2 (maybe 3) we like to say the dx at the end is just some notation (a reminder of what you are doing) which is integrating with respect to x. However, in calc 1 dy (or dwhatever) is typically defined as an approximation which is not the same thing that's happening when you put the dx at the end of your integral. There are a lot of really good KZbin videos which explain differential forms really well so if you really like math I suggest you search for them.

[@rishikothari340]

Integration is summation using rectangles of infinitely small width?

[@guitarttimman]

Not exactly, but that's the right idea.

[@guitarttimman]

Repeat after me: Infinitesimal. IN FA NA TESS A MUL

[@helloitsme7553]

The integral is what the riemann sum converges to when taking the rectangles smaller and smaller. So in a way you could say that, but be careful: actually summing 0 infinitely many times is undefined, this is taking the limit which is defined

[@guitarttimman]

@@helloitsme7553 Actually, the more conservative way is to find largest sub-interval, find the mid-point, plug the mid point into the function, multiply them and then take the limit to find the area under the bounded region of the cuve. In three space, use the largest diagonal of the subinterval. Oh by hey! What do I know right? LOL

[@thaioanquoc6129]

i have studied integral for a year and still confused about what is a dx until I found your video. thanks

[@ronojoarder8850]

my teachers never teach like this way. salute u

[@vic123]

The dx notation is super clever in differential equations too imo. It lets you treat them as if they were just regular variables, making the notation much more intuitive than it originally is in calc 1, for example

[@ibrahimawadallah]

Congratulations on finally hitting the 100,000 subscribers! I've been waiting for weeks haha! Your videos are amazing! Keep up the great work!

[@kristakingmath]

+Ibrahim Awadallah Aww thank you so much! I'm so excited for the next 100K!

[@aaliyahvarona5409]

Wow, this video may be old, but it still helps. My teacher rushed through the explanation that the only thing I understood was limit sense from what he said, but now it's clear what we are talking about in terms of limit sense. Thank you, ma'am!

[@p.c2750]

A step closer to 100% understanding of these things. Thank you

[@kristakingmath]

You're very welcome, Pearl! :)

[@shawnlove7417]

Beautifully explained! As a student applying to be a physics learning assistant, I cannot wait to pass this video along to my peers.

[@kristakingmath]

Thank you so much, Shawn! Please do share with anyone who could use it! :D

[@integralmath]

For the latter explanation in the change from delta x to dx, I think it would have been useful to make explicit that the transition comes because of the limit. As the width of the delta x becomes infinitely smaller because infinitely many (rectangles, trapezoids, squirrels [don't judge me; I regret nothing], whatever) are used to improve the approximation, the switch from delta x to dx occurs.

[@ralfanari8854]

+The Justicar or you could just say that an integral represents the area/space/however deep you may want to go and dx represents the axis you are multiplying your function with to get the spatial dimension

[@integralmath]

+Ralf Anari that wouldn't be so though.

[@ralfanari8854]

The Justicar elaborate, I mean in mathemathical proofs you cant say "it symbolizes area, forget about it", but dx, delta x, delta=change in the x axis, so you are multiplying your function relative to the change of whatever axis you are integrating at the given moment, measuring the change of the function you integrate relative to the constant change of your axis.

[@integralmath]

+Ralf Anari you couldn't truthfully say that in normal conversation either. "it" in this case is "dx". And "dx" simply does not symbolize area. Further, "dx" doesn't simply represent change along the x-axis."dx" represents a specific type of change, its particular type of change only becomes precise because of the limit. "x" is the axis. "dx" represents something else. Or, if it doesn't, then you could say that dx = x. But dx doesn't equal x; d, whatever it is, must bring something else to the table that distinguishes x from dx. And what it brings is the notion of a particular type of change which is coherent only when the notion of the limit exists in conjunction with it.

[@ralfanari8854]

The Justicar I never said it symbolizes area, it symbolizes you calculating area relative to the axis x, every integral has one constant axis and one changing axis(the function you are integrating), you average out the change in the function relative to the change of your axis and you get....wait for it....the area edit:I might have misworded it at my first comment, at the end I said dx represents teh axis, i shouldve clarified that it represents the start and the end of the points on the axis you are integrating in case you are not integrating from infinity to infinity

[@bjornfeuerbacher5514]

2. You don't need multivariable calculus in order to see why the dx is important; it already becomes apparent why it's important when one is teaching integration by substitution properly. 3. This should _always_ be mentioned _first_, since that's exactly the _historical_ reason why that notation was actually developed! All other explanations only came afterwards and are essentially unimportant.

[@Inspirational_Homes]

I have been studying origin of derivatives,and understood them completly with concepts,and from some videos i actually got it very clearly what intergration and diffrentiation really is,its not a diificult thing to understand rather people make and sound like its very hard to understand.Though i really liked your explanation for the derivative multiplying with the integral.Thanks soo much Ma'am :-)

[@Miss_blizzard187]

I'm in tears! You're great ma'am, keep going 👍🏻

[@Albert-ct6tt]

Great explanation, very straight and rigorous at the same time. My compliments!

[@kristakingmath]

Thank you so much, Albert! :)

[@masonr1666]

It's also worth to note that when integration occurs and you are taking an area, typically you are finding the area between two locations, so by having the dx you know the starting point, and the ending point of that specific range if the function/curve/graph.

[@mechwarreir2]

Integrals don't always have the "dx" necessarily. In exterior/geometric calculus there are different ways of representing differential forms by observing the cross product (called wedge product) of differential basis vectors.

[@thetrickster42]

mechwarreir2 I'm a big fan of dropping the wedge in the wedge product, and just writing the differential forms side by side.

[@ZipplyZane]

While I agree dx is infinitely small, I think it would be appropriate to point out that it is the limit as ∆x (the width of the rectangles) approaches 0. If the function is continuous, the limit at x→a of f(x) = f(a)

[@tonato17]

Thank you very much. I've been trying to fully understand this concept and its relation to integrals and I still find it hard to grasp. I asked a teacher bu he just told me that the dx it's an expression, which I don't agree. I understand that dy/dx = f'(x) because the change that dy experiences when dx is too small (tends to 0) is what gives the slope at any particular x value. which is the definition of a derivative. I understand that from dy/dx = f'(x) I can also write dy=f'(x) . dx. which is the formula we use to find the integral. which makes me think two things... 1) is dy the original formula then? because when I do f'(x) . dx what I find is the original formula. If so , how is that? I struggle to see it. 2) If dx is extremely small, if I multiply something for an extremely small value, wouldn't that give 0 as a result? I mean, why is it that I discard the dx when I find the primitive of f'(x)?if it's so small the result would be close to 0 so I don't understand why we just find the primite and discard the dx. This concept is driving me a bit crazy. Thank you!!

[@N1colastoledo]

The last explanation was really great, but I've a question: that interpretation only applies to the definite integral? I mean, the indefinite integrals or antiderivatives aren't just the opposite of derivatives? I would appreciate your help

[@The053199]

They are. The symbol was "misused" at first for the indefinite integral and people just went with it for the definite integral

[@srikanthtupurani6316]

nice explanation. most of the people are confused about it. it is a notation. in case of pure math there are many things which are confusing example when someone says let us consider the measure (X^2+1) dx. this is confusing . in case of differential forms we use the symbols dx1,dx2,.... many people are confused about this notation. here dx1 is not differential elemment of x1.

[@shivamtiwari3679]

You make calculus interesting and easy to digest. You r great !

[@kristakingmath]

Thank you so much, Shivam! :D

[@HDitzzDH]

So understanding the definition of the integral (Riemann integral) you will understand why "dx" is there in the first place, when you have an limit of an infinite sum you can naturally translate it into an integral?

[@soydavidsolis]

I was just waiting for the last answer. Great video.

[@billusandda5845]

This channel is so underrated i😭😭😭

[@kristakingmath]

Thank you so much, Nargish! :)

[@dijonstreak]

thank you SO much...i finally unserstand why that dx is at the end of every integration i have seen....thank you for a great demo..!!

[@alphabetdot6613]

That's why I paid KZbin For Remove Ads, & That's why KZbin is Heaven for Self-Study, before watching this video i don't know why i's even using🤯amazing vid ❤

[@dijonstreak]

THAT was AWESOME !! Thanks to. YOU i FINALLY understand the dx notation...thank YOU. SO much for avery awesome explanation i am finally in Calculus. thank. YOU. again !!!!!!

[@Carrymejane]

I haven't learn calculus yet but i understand a whole concept

[@kristakingmath]

I’m so glad it made sense!! :)

[@classifiesconfidential4330]

10 second answer, to symbolically represent the summation of infinite rectangles with area of f(x) times “dx” a small change in x.

[@bobprior6661]

Thank you. Great explanation, especially the relationship between Delta x and dx. Another reason the integral has dx is the fact that the integral represents the antiderivative. What I mean by this is, when we first learn the rules of derivatives, we are taking the derivative of a function in terms of x; it is written as an equation, and we know, "Whatever you do to one side, you must do to the other." (Another way to say this is, "However you operate on one side, you must operate the same on the other side.") So, when we take the derivative, we take the derivative of each side of the equation simultaneously: y = f(x) begets y' = f '(x). We know this also as dy/dx = f '(x), and we can multiply each side by dx to get dy = f '(x)dx. From this equation, we can now use the operation "Integral" INT[ ] on each side, giving us INT[dy] = INT[f '(x)dx]. The left side is really INT[1dy] and the antiderivative (in terms of y) is just y (let's not worry about +C at this moment). The right side antiderivative is just f(x), giving us what we started with, y = f(x) (okay ... + C). Again, thank you for this quick, important video.

[@toby9999]

Seems as though your explanation was heading toward an answer to a question I posted but not quite. I don't understand what the dx means in the front of an integral. If it means the derivative then why would we need it? Why should we care when we're actually interested in the integral? Why is it even relevant? For instance, if I want to integrate 2x, I just write x^2 + C. The dx played no role.

[@vishalbhatia70]

In Integral Calculus we do Summation of Continuous extremely small particles In Statistics we used the Capital Sigma symbol to do Summation of Discrete Variables - which may be small or even bigger - such as say 1+2 etc etc

[@arvindian]

can u explain d/dx too why division why not any other operation?

[@kristakingmath]

+arv synth Great idea, I'll add that to my list!

[@denizn.tastan6847]

arvind d Because Derivative of a function is the slope of the function at any given point. Usually on straight lines formula of slope is change in y over change in x. m=y/x or m=Δy/Δx. Since y=f(x) d/dx f(x) also means d.f(x)/dx

[@arvindian]

thank u

[@AbiRizky]

It makes more sense in physics to be honest, as to why it is division. like, dd/dt = v kinda makes more sense.

[@netoskin]

Search for the slope formula which is m = y2 - y1/x2-x1 so when we want the slope to be touching a single point in the curve(that means our two points to get the slope must be sooo close going to zero) we say is the difference of y / difference of x so dy/dx

[@8centstrading]

I haven't watched any of your videos since my freshman year of college (2012). Regardless, I'm still subscribed and excited to see how much your channel has grown

[@kristakingmath]

+Richard McGrath Wow, thank you so much Richard! I appreciate the support! :D

[@david04-i9]

Thanks a lot! YT just recommended your videos to me and I love the way you explain things!

[@finnmcmisslefanchannel-pt3xu]

THANK YOU, this makes so much more sense now

[@kristakingmath]

Oh good! So glad it helped! :D

[@feriacientifica6139]

You are getting better and better, many (...or much ???) thanks from Santiago of Chile !!

[@kristakingmath]

Aw thanks Feria!

[@byteaesx1373]

A trillion thanks for your time and effort! I wish you the very best Krista.

[@kristakingmath]

Thank you so much! :D

[@vishalbhatia70]

dx stands for INFINITESIMALLY SMALL PARTICLES OF X WHICH HAVE ACTUALLY BEEN DIFFERENTIATED THE INTEGRAL SIGN COMES FROM THE LATIN LONG S WHICH STANDS FOR SUMMA OR SHORT FOR SUMMATION Gotfried Leibnitz used these notations for the 1st time !! Hope the above helps

[@klausgiesselmann4075]

vi - "Gotfried Leibnitz used these..." Don't goof around with names! His name is Gottfried Leibniz!

[@dribblelin2481]

I wish I had a math teacher like you!

[@Kokurorokuko]

by infinitely summing antiderivatives you get the sum of all possible function values at every point (or sum of heights of rectangles). But to get the area under the curve you need to multuply it by the dx (which is the base of the rectangles).

[@TheCarkolum]

This is the only right ASMR

[@kristakingmath]

I love ASMR! :)

[@albopoker8333]

very well explained ! finally got d answer to why do we actually need to integrate ! thanks !

[@kristakingmath]

:D

[@stuart5960]

Thank you so much, Krista!

[@BaryLevi]

in measure theory we use the d(something) to notate with respect to which measure to integrate, such as dμ, dλ . This way it also does not feel redundant anymore.

[@militantpacifist4087]

I always saw the dx as a difference in x's since after integrating you had to subtract between them. You're almost to 100K subscribers by the way.

[@kristakingmath]

+Militant Pacifist That's a great way of thinking about it! And yes, just crossed 100K! :D

[@BigDBrian]

although it doesn't negatively impact you, that insight is incorrect.

[@anonymous-zy7pv]

One of the best teachers in the world. I subscribed for you❤

[@kristakingmath]

Thank you so much for subbing! :)

[@SadBulldog]

*congratgulations you have successfully made it to my ASMR playlist*

[@adityabhardwaj3495]

Your voice cured my fever 😌😊

[@markknight5178]

Puts my professors explanation to shame...

[@sladkyhleb6688]

You are right, probably. But when we are supposed to find out an area under a curve, we do not count any area of each rectangle, instead of this we shoud find аntiderivative of a function. After this we subtract values, which are limits of the integration, using аntiderivative of a function. Sorry for grammatical mistakes, I am from Russia.

[@irlandadelrazo2661]

Mejor explicación no hay! Excelente!

[@georgepolasky9809]

Such a very wonderful explanation, yet again, Krista. Thank you very much.

[@kristakingmath]

You are so welcome, George! :D

[@andybaldman]

Your voice makes men fall in love with you.

[@Ben-bf5lf]

video stars at 0:32

[@aventurileluipetre]

Thank you

[@siveenahmed7195]

Girl u're amazing trust meee❤️❤️❤️

[@jason-xn9le]

When taking the limit of the reiman sum I would just think of it as f(x) x dx or “height x width” I think dx is always there because as “n” approaches infinity, the width of delta x infinitly shrinks as the number of rectangles under the curve infinitely increase in number. Think of it as the b-a/♾

[@HeyKevinYT]

I remember being picky with my classmates and yell at them everytime they don’t write the dx 😂

[@unknownnepali772]

If dx is only notation and it is used only to identify which term to integrate then, why we convert dx in terms of d(theta) during integration in trignometric substitution....yeahh you might say since we are integrating theta so we convert dx in terms of theta but my question is if dx is only for notation for understanding which part to integrate then dx should be completely useless but in trigonometric substitution we use dx in different way......and if dx is just notation then in trig substitution why can't we just replace dx with theta only but that's wrong...there might be big connection. we can pretty much easily recognise which terms to integrate...i think we don't need just notation........i am currently studying these minor things to fully understand mathematics but i lack people who can support me n i can't find my all questions in internet and these type of questions are neither available in text books...i hope i could take tution with a mathematician😅😁😉

[@dard4642]

I learned to think of dx as "delta of the x values" but such an infinitesimally small change in x that it can only be defined as "not no change in x." I have a BS in Mathematics and one of the best classes I ever took was a low-level class where we used no textbook and didn't do a whole lot of calculations -- we did a lot of writing. We had to write in paragraphs how we would go about solving a particular problem. It wasn't a proofs class, and we weren't using theorems. Our professor's goal was to get us to think, not to calculate. I can only imagine how much his TA hated him because each student's solution was unique. Some people would solve a problem in two paragraphs, other people took two pages; what mattered is that you arrived at the destination. Anyway, that's just a very long-winded way of me telling you that I appreciate your explanation.

[@kristakingmath]

That sounds like a math class everybody should take! I love the idea. I'm so glad you enjoyed the video! :)

[@jiewang6449]

Great explanation! Thanks!

Thank you for your explanation. Now that's clear to me.

[@ahmedengineer5778]

thanks for your clarity

[@fikret8422]

thank you ı know that why we write dx but even so you told great actually these type of videos can help us to learn about that topic more clearly

[@kristakingmath]

I'm so glad you liked it!

[@georgemissailidis7581]

0:54 so dx is just like a full stop, and the rets of the integral equation is your sentence, and your sentence starts with a big capital S :)

[@johnpro2847]

this has bothered me for years..thanks

[@kristakingmath]

You’re welcome, I’m so glad this helped clarify!! ☺️

[@deadjunior258]

Let's say: dy/dx = 2 If I move the dx to the RHS, I get: dy = 2*dx If I integrate both sides, it will look like this: int. dy = int. 2 dx int. 1 dy = int. 2 dx dy = 1*dy If you complete the integration, you get: y = 2x + c That's what I learnt in school. Correct me if I'm wrong.

[@nikibl8]

How do we think about the dx in an integral when the differential is viewed as a function df(x):R →R denoted df(x)(h)=y'h?

[@michaelnichxls]

I have no idea how you made me understand everything you said

[@RaymundoGabriel]

Thank you!! You make it so easy to understand!

[@kristakingmath]

You're welcome, I'm so glad it helped! :D

[@guitarttimman]

If you view dx as being a width, then you'll get it.

[@justtheletterV274]

If we integrate dy/dx in order to get y, we first need to multiply it by dx in order to get dy, and then integrating dy will get just y.

[@carultch]

We do. It's only half the story to say that we integrate dy/dx in order to get y.

[@kendout]

Love your voice and your vid really take me to the point.

[@phatp4206]

dx = with the respect to a tiny change in x OR dx = delta x

[@jaylakhani5568]

Excellent video, I was happy at the end - thanks

[@kristakingmath]

You're welcome, Jay!

[@leZigoute]

Thank you very much for the explanations.

[@jimcar53]

Thanks for the explanation

[@ruqqiyanobat7270]

Thanks for explaining this ...

[@micah2501]

This is a great explanation. Thank you!

[@kristakingmath]

You're welcome, Runnermif, glad it was helpful! :)

[@feleous5081]

I Think instead of dx= infinitely small distance between two x values its rather, the distance between to x values that approaches zero ( you could phrase this as a limit).

[@durragas4671]

quick and to the point! ty

[@kristakingmath]

You're welcome, Durragas! :)

[@motivationformuslims1214]

Thanks for this very useful video