This is one of those videos that make maths look hard and scare students away from maths. Instead of telling people supposedly the "types of integrals", tell them it's the same basic idea but it is profoundly useful in many ways (your "type of integrals").
@@s.rehman2.0 well, as we said in the beginning of the video, we did not intend to explain in detail any of the integrals in this video, but instead just literally show the list. In the next video in the channel we will explain the RS integral. I’m sure after that you will be much less scared of the math 😉 don’t worry, you'll be fine 😌
@@Frdnnd thanks for the nice words!! It is ok, something I’ve learned in all these years is that when we see topics that are more advanced than our current knowledge we tend to be intimidated, but what we fail to realize is that if the topic didn’t challenge your brain it means that you are not learning. So, keep on trying to understand topics that are “too advanced” for you and eventually you will get comfortable with them, aka you’ll learn 😎
Although knowing these concepts and where to use them is really useful, I wouldn't categorize them as "types of integrals", some where legit different definitions of integral, but most of them where just famous/useful applications/transformations dealing with Lebesgue integrals. And since the Lebesgue integral is one of the most important ones talking more in depth about it would be really cool, specially the theorems of representation which indeed confirms that the Lebesgue integral is the correct generalization of the Riemann Integral
@@aangulog you are correct to some extent. I would even go further and notice that some of them are generalizations of one another, or particular cases of one another. But the point is that the techniques used to solve each of these integrals are different and their geometrical interpretations are different (even though all of them come down to limits of sums at the end of the day), so it does make sense to put them in different buckets, just as it’s currently done by the math community nowadays, and that’s why they have their own names and particular notations.
@@dibeos Indeed, the different interpretations that they can have is the real amazing part about all these. Thanks for the response, Like your content ❤
I would add product integral, where the "small pieces" usually have values close to unity instead of zero, and are multiplied instead of summed. For ordinary numbers, we have P∫ f(x)^dx = exp (∫ ln f(x) dx), but when the function takes values in a non-commutative algebraic structure (e.g. a group), then things become more interesting. The solution of a time-dependent Schrödinger equation can be written as a product integral.
@@fdileo I’m preparing the video, it’s just that its visualization is tricky so making it is taking me a lot of time. Anyway, it will be very clear because I’ll even will show 3 concrete examples of it. It’s coming soon!!! 😎
Also on a separate note here… the Ito integral is a member of a broad class of stochastic integrals. There many more stochastic integrals, Ito is the most common thats rises naturally when building stochastic DE. Looking forward to your take on RS integral. There’s a nice paper from 1989 going about geometry behind RS integral. Essentially you end up calculating areas of projected fx over the weight function induced by g(x) and its derivative. After that sums and integral fused in a single object on my head. 😅
@@rick4135 exactly!!! I think it is not difficult to understand conceptually what is the area that we are calculating with the RS integral, but I’m honest suffering to represent geometrically in 3D such that it will be clear hahaha I need to work on it!!! 😝
@@dibeos I've been an integration enthusiast for as long as I can remember, ever since Calculus 1 (a long time ago). However, my understanding of its mathematical interpretation has evolved significantly over time. I’m not sure if your take on the geometrical interpretation aligns with mine, but here’s how I see it: If you plot in 3D: f(x) = x g(x) = x^2 Let's say x is in the unit interval. Then, x with respect to dx moves at the same rate. The projection on the xz -plane will be z = x , as expected. Now, x with respect to dg(x): In the context of g(x), the changes in x appear slower. In fact, g(x) perceives x as sqrt{x} when x itself sees x^2 in the x-g plane When projecting onto the zg plane, the result is z = \sqrt{x} . To find the area under this curve, you just need to adjust the limits of integration accordingly. Since g(x) is “faster,” the limits of integration also need to be adjusted. This is essentially the inverse of what typical textbooks describe when calculating the Riemann-Stieltjes (RS) integral. Notice here that we have: (latex here) \[ \int x \, dg(x) = \int x \cdot (2x) \, dx = \frac{2}{3} \] This is the same value as: \[ \int \sqrt{x} \, dx = \frac{2}{3} \] both over the unit interval. This means that x behaves like sqrt{x} when the differential rate is squared. If you plot these shapes in 3D, the differences in the xz -planes and gz -planes become very clear. This could serve as a precursor to the Radon-Nikodym theorem in measure theory. In statistics and probability theory, it represents how we evaluate and transform probability spaces for various random variables. In this sense, you can generate all random variables by simply choosing a random point in the unit interval. This essentially makes the uniform distribution the most fundamental, as it allows the generation of all other distributions starting from it.
What about the Mcshane Integral? I love it as it can be thought of as Generalizing the Gauge Integral in a way, and relating to the Lebegue integral in a way unifying the theory of integration. This is not the best explanation but you should check it out!
Love from india sir and mam. 1st time i have watch your video on Group theory which is Galois theory. I impressed you and mam. Thank you for giving us valuable information.
@@hazimahmed8713 A double integral refers to integrating over a region in 2D space, whereas a surface integral is more general, used to integrate over a surface in 3D space. A similar difference exists between a triple integral (over a volume in 3D space) and a volume integral (specifically over the volume of a solid object). It is a generalization. For example, let’s say you want to find the amount of water collected on a flat field after rain. A double integral would help you calculate the total amount based on how much water is in the area of the field (considering the certain height of water at each point, but you get the point). Now, think about a curved object, like a little mountain. To find the total amount of rain on the hill's surface, you would use a surface integral. It takes into account not only the area but the curvature of the surface (embedded in 3D). For a triple integral, imagine you're calculating the total mass of a 3D object, like a solid cube, where the density varies throughout the volume. A volume integral would help you find the total mass based on this varying density. But if you wanted to calculate the total mass inside a specific shape or region (like inside the cube), you'd use the volume integral for that region.
@@jammasoundthanks!! Well, let us know what difficulties you’re finding in learning about the Riemann integral. Maybe we can make a video about it in order to help you 😄
@@filipnagy3535 Hm… yeah, good point. Integrals of differential forms on manifolds are indeed important, and they should be considered their own type. But they generalize concepts like line, surface, and volume integrals. I focused on more elementary types of integrals in this video, but integrals on manifolds definitely deserve a mention in a broader discussion. Well, as I said in the video: this list is not exhaustive, but it is an extensive one. Thanks for the comment 😎
@@InputOutput-b2l yessss we will love to do it!!! So far in the list we have (in this order): (1) RS integral ; (2) Bochner integral ; (3) Feynman integral
@@Satisfiyingvideo-uu9pw There isn't a simple closed-form formula for the product of numbers in the sequence 1, 3, 5, ..., 2n-1. The product can be expressed as: P(n) = 1 × 3 × 5 × ... × (2n-1). This is also: P(n) = ∏(2k-1) from k=1 to n. However, unlike the sum of the first n odd numbers (which equals n^2), there isn't a basic formula for this product. Does it answer your question?
This video is awesome!!! Didn’t knew s couple here… check the product integral. Quick application: survival analysis for estimating the survival function. A data driven analog is related to the kaplan meier estimator of survival function of a non negative random variable
@@dibeos I’m referring to the product integral, is not part of you list. swap sums for product in the integral definition. Some summaries below: 1. **Kaplan-Meier Estimator**: A non-parametric method to estimate survival functions with censored data. 2. **Multiplicative Updates**: Survival probabilities are updated multiplicatively at each event time, based on the number of events and individuals at risk. 3. **Estimator Formula**: \(\hat{S}(t) = \prod_{t_i \leq t} \left( 1 - \frac{d_i}{n_i} ight)\). 4. **Product Integral Relation**: The estimator follows a product integral structure, multiplying small survival decrements over time. 5. **Continuous-Time Form**: The product integral form is \(\hat{S}(t) = \prod_0^t \left( 1 - \lambda(s) \, ds ight)\), where \( \lambda(s) \) is the hazard rate. 6. **Hazard Rate Definition (in terms of density)**: The hazard rate \( \lambda(t) \) is given by: \[ \lambda(t) = \frac{f(t)}{S(t)} \] Where: - \( f(t) \) is the **probability density function (PDF)**, representing the rate at which events occur at time \( t \). - \( S(t) \) is the **survival function**, which is the probability of surviving beyond time \( t \) (i.e., \( S(t) = 1 - F(t) \), where \( F(t) \) is the cumulative distribution function (CDF)).
@@sphakamisozondi yeah, it is quite interesting, isn’t it?! Of course, one can integrate many types of functions, but those in the video are different types of integrals. Just saying, because people can get these terms mixed up
This is one of those videos that make maths look hard and scare students away from maths. Instead of telling people supposedly the "types of integrals", tell them it's the same basic idea but it is profoundly useful in many ways (your "type of integrals").
@@s.rehman2.0 well, as we said in the beginning of the video, we did not intend to explain in detail any of the integrals in this video, but instead just literally show the list. In the next video in the channel we will explain the RS integral. I’m sure after that you will be much less scared of the math 😉 don’t worry, you'll be fine 😌
I'm 20 and haven't even finished* algebra yet; I'm not sure if i belong here, but your channel is somehow still captivating to an amateur like me.
@@Frdnnd thanks for the nice words!! It is ok, something I’ve learned in all these years is that when we see topics that are more advanced than our current knowledge we tend to be intimidated, but what we fail to realize is that if the topic didn’t challenge your brain it means that you are not learning. So, keep on trying to understand topics that are “too advanced” for you and eventually you will get comfortable with them, aka you’ll learn 😎
Although knowing these concepts and where to use them is really useful, I wouldn't categorize them as "types of integrals", some where legit different definitions of integral, but most of them where just famous/useful applications/transformations dealing with Lebesgue integrals.
And since the Lebesgue integral is one of the most important ones talking more in depth about it would be really cool, specially the theorems of representation which indeed confirms that the Lebesgue integral is the correct generalization of the Riemann Integral
@@aangulog you are correct to some extent. I would even go further and notice that some of them are generalizations of one another, or particular cases of one another. But the point is that the techniques used to solve each of these integrals are different and their geometrical interpretations are different (even though all of them come down to limits of sums at the end of the day), so it does make sense to put them in different buckets, just as it’s currently done by the math community nowadays, and that’s why they have their own names and particular notations.
@@dibeos Indeed, the different interpretations that they can have is the real amazing part about all these. Thanks for the response, Like your content ❤
@@aangulog thank you Andrés 😎
I would add product integral, where the "small pieces" usually have values close to unity instead of zero, and are multiplied instead of summed.
For ordinary numbers, we have P∫ f(x)^dx = exp (∫ ln f(x) dx), but when the function takes values in a non-commutative algebraic structure (e.g. a group), then things become more interesting. The solution of a time-dependent Schrödinger equation can be written as a product integral.
@@md2perpe interesting!!! I didn’t know! That’s very cool. I will search more about it 😎
I knew around 21/22 of them. I'm looking forward to watching the Riemann-Stieltjes integral video
@@fdileo I’m preparing the video, it’s just that its visualization is tricky so making it is taking me a lot of time. Anyway, it will be very clear because I’ll even will show 3 concrete examples of it. It’s coming soon!!! 😎
Also on a separate note here… the Ito integral is a member of a broad class of stochastic integrals.
There many more stochastic integrals, Ito is the most common thats rises naturally when building stochastic DE.
Looking forward to your take on RS integral. There’s a nice paper from 1989 going about geometry behind RS integral.
Essentially you end up calculating areas of projected fx over the weight function induced by g(x) and its derivative.
After that sums and integral fused in a single object on my head. 😅
@@rick4135 exactly!!! I think it is not difficult to understand conceptually what is the area that we are calculating with the RS integral, but I’m honest suffering to represent geometrically in 3D such that it will be clear hahaha I need to work on it!!! 😝
@@dibeos
I've been an integration enthusiast for as long as I can remember, ever since Calculus 1 (a long time ago).
However, my understanding of its mathematical interpretation has evolved significantly over time.
I’m not sure if your take on the geometrical interpretation aligns with mine, but here’s how I see it:
If you plot in 3D:
f(x) = x
g(x) = x^2
Let's say x is in the unit interval.
Then, x with respect to dx moves at the same rate.
The projection on the xz -plane will be z = x , as expected.
Now, x with respect to dg(x): In the context of g(x), the changes in x appear slower. In fact, g(x) perceives x as sqrt{x} when x itself sees x^2 in the x-g plane
When projecting onto the zg plane, the result is z = \sqrt{x} .
To find the area under this curve, you just need to adjust the limits of integration accordingly.
Since g(x) is “faster,” the limits of integration also need to be adjusted. This is essentially the inverse of what typical textbooks describe when calculating the Riemann-Stieltjes (RS) integral.
Notice here that we have: (latex here)
\[
\int x \, dg(x) = \int x \cdot (2x) \, dx = \frac{2}{3}
\]
This is the same value as:
\[
\int \sqrt{x} \, dx = \frac{2}{3}
\]
both over the unit interval.
This means that x behaves like sqrt{x} when the differential rate is squared. If you plot these shapes in 3D, the differences in the xz -planes and gz -planes become very clear.
This could serve as a precursor to the Radon-Nikodym theorem in measure theory. In statistics and probability theory, it represents how we evaluate and transform probability spaces for various random variables.
In this sense, you can generate all random variables by simply choosing a random point in the unit interval. This essentially makes the uniform distribution the most fundamental, as it allows the generation of all other distributions starting from it.
What about the Mcshane Integral? I love it as it can be thought of as Generalizing the Gauge Integral in a way, and relating to the Lebegue integral in a way unifying the theory of integration. This is not the best explanation but you should check it out!
@@daxplatiro1668 yeah, we missed this one… anyway I’m gonna search about it right now and add to our list of video ideas 😎
love your work
@@rewixx69420 thank you!!! Let us know what kind of content you’d like to see in the channel 😎
@@dibeos im trying to understand puiseux series its real hard to do and i dont know how it cold be derived
@@rewixx69420 we will make a video about it
@@dibeos thanks
Love from india sir and mam.
1st time i have watch your video on Group theory which is Galois theory. I impressed you and mam.
Thank you for giving us valuable information.
@@OpPhilo03 thank you!!! We are happy you enjoyed the video and learned from it 😎
What is the difference between a surface integral and a double integral? Same doubt I have for volume integral and triple integral.
@@hazimahmed8713 A double integral refers to integrating over a region in 2D space, whereas a surface integral is more general, used to integrate over a surface in 3D space. A similar difference exists between a triple integral (over a volume in 3D space) and a volume integral (specifically over the volume of a solid object). It is a generalization.
For example, let’s say you want to find the amount of water collected on a flat field after rain. A double integral would help you calculate the total amount based on how much water is in the area of the field (considering the certain height of water at each point, but you get the point).
Now, think about a curved object, like a little mountain. To find the total amount of rain on the hill's surface, you would use a surface integral. It takes into account not only the area but the curvature of the surface (embedded in 3D).
For a triple integral, imagine you're calculating the total mass of a 3D object, like a solid cube, where the density varies throughout the volume. A volume integral would help you find the total mass based on this varying density.
But if you wanted to calculate the total mass inside a specific shape or region (like inside the cube), you'd use the volume integral for that region.
not sure if the 3 types of product integral have geometric interpretations, but I didn't see them on the list either way
cool video. Waiting for the next one
Yay, new video)
And like I know 12/33
@@SobTim-eu3xu that’s awesome!!! Soon we will publish about RS integral
@@dibeos oh, cool)
Nice overview, I'm still learning on the Riemann integral 😃
@@jammasoundthanks!! Well, let us know what difficulties you’re finding in learning about the Riemann integral. Maybe we can make a video about it in order to help you 😄
what about integral of forms on a manifold?
@@filipnagy3535 Hm… yeah, good point. Integrals of differential forms on manifolds are indeed important, and they should be considered their own type. But they generalize concepts like line, surface, and volume integrals. I focused on more elementary types of integrals in this video, but integrals on manifolds definitely deserve a mention in a broader discussion. Well, as I said in the video: this list is not exhaustive, but it is an extensive one. Thanks for the comment 😎
Cam you do a video about the feyman integral??
@@InputOutput-b2l yessss we will love to do it!!! So far in the list we have (in this order): (1) RS integral ; (2) Bochner integral ; (3) Feynman integral
@@dibeos can't wait for the next video🙇
Please make a video about the Bochner integral! I've come across it during research and I need to better understand it. Thanks a lot. 💖
@@mahmoudhabib95 I will!!! I’ll add to my list of next videos right now 😎
@@dibeos
You’re amazing. Thank you! 😊
is there any formula for pi product. Like i want to multiply 1,3,5,.....2n-1 .is there for sequence like an+b here a and b is constant
@@Satisfiyingvideo-uu9pw There isn't a simple closed-form formula for the product of numbers in the sequence 1, 3, 5, ..., 2n-1. The product can be expressed as: P(n) = 1 × 3 × 5 × ... × (2n-1). This is also: P(n) = ∏(2k-1) from k=1 to n. However, unlike the sum of the first n odd numbers (which equals n^2), there isn't a basic formula for this product. Does it answer your question?
@@dibeoscan i get formula for approximation
@@Satisfiyingvideo-uu9pw 1,3,5,.....2n-1=(1*2*3*4*5*...*(2n-1))/(2*4*6*8*...*(2n))=(2n-1)!/2(n!)
This video is awesome!!! Didn’t knew s couple here… check the product integral.
Quick application: survival analysis for estimating the survival function. A data driven analog is related to the kaplan meier estimator of survival function of a non negative random variable
@@rick4135 it seems to be very interesting! Can you expand it, please? Also, I didn’t understand which integral of the video you referring to
@@dibeos I’m referring to the product integral, is not part of you list.
swap sums for product in the integral definition.
Some summaries below:
1. **Kaplan-Meier Estimator**: A non-parametric method to estimate survival functions with censored data.
2. **Multiplicative Updates**: Survival probabilities are updated multiplicatively at each event time, based on the number of events and individuals at risk.
3. **Estimator Formula**: \(\hat{S}(t) = \prod_{t_i \leq t} \left( 1 - \frac{d_i}{n_i}
ight)\).
4. **Product Integral Relation**: The estimator follows a product integral structure, multiplying small survival decrements over time.
5. **Continuous-Time Form**: The product integral form is \(\hat{S}(t) = \prod_0^t \left( 1 - \lambda(s) \, ds
ight)\), where \( \lambda(s) \) is the hazard rate.
6. **Hazard Rate Definition (in terms of density)**: The hazard rate \( \lambda(t) \) is given by:
\[
\lambda(t) = \frac{f(t)}{S(t)}
\]
Where:
- \( f(t) \) is the **probability density function (PDF)**, representing the rate at which events occur at time \( t \).
- \( S(t) \) is the **survival function**, which is the probability of surviving beyond time \( t \) (i.e., \( S(t) = 1 - F(t) \), where \( F(t) \) is the cumulative distribution function (CDF)).
I never knew there are different types of intergrals. I thought there were two, 😅
@@sphakamisozondi yeah, it is quite interesting, isn’t it?! Of course, one can integrate many types of functions, but those in the video are different types of integrals. Just saying, because people can get these terms mixed up