What is the concept of distance as an integral in Feynman lectures on physics?

[rudransh verma]

From Feynman lectures on physics: https://www.feynmanlectures.caltech.edu/I_08.html
Page 8-7 (Ch 8 Motion)
“ To be more precise, it is the sum of the velocity at a certain time, let us say the ith time, multiplied by deltat.
$s=\sum_{i} v({t_i})\Delta t$”

Now I suppose ${t_i}$ is some time instant in $\Delta t$ interval.
But to be more accurate we need to decrease the $\Delta t$ so that velocity doesn’t change in that interval. Taking limit we get
$s= \lim_{\Delta t \rightarrow 0} \sum_{i}v({t_i})\Delta t$.
I don’t know how to take limit when summation is included. Help !

 

[wrobel]

First study the Riemann integral by a math textbook.
It is not a good idea to study math by physics textbooks

 

[rudransh verma]

First study the Riemann integral by a math textbook.
It is not a good idea to study math by physics textbooks

I am trying to understand integral from scratch. That’s why I am asking what is the limit of summation?
Please read the module on page 8-7 in book.

 

[Ibix]

I am trying to understand integral from scratch. That’s why I am asking what is the limit of summation?

Feynman doesn't explain this. At least in this context, he is content to note that other people have shown that the limit is an integral. As @wrobel says, the person who first wrote this proof was Riemann. You can either accept that he did it and not worry about it, or you can find a maths textbook and look up the Riemann integral.

 

[rudransh verma]

You can either accept that he did it and not worry about it, or you can find a maths textbook and look up the Riemann integral.

In our textbook(NCERT) in India we are not taught about Riemann integrals. We are by the way taught that derivatives are nothing but limits. But not the integrals. Can you suggest any book where I can look up the procedure of getting to the final integration line that is the very simple integrations.

 

[jbriggs444]

In our textbook(NCERT) in India we are not taught about Riemann integrals. We are by the way taught that derivatives are nothing but limits. But not the integrals. Can you suggest any book where I can look up the procedure of getting to the final integration line that is the very simple integrations.

I went through college as a math major. Through several years of education involving integrals, differentials, differential equations and such there was only one occasion in one class where the formal definition of a Riemann integral was even attempted. Possibly five or ten minutes. As I recall, the limit is taken of the "mesh".

I did not take much away from that episode other than the idea of a mesh. However, that was enough. From that starting point, one can deduce pretty much how the definition has to be formulated.

The idea is that you partition the interval into pieces, each of length smaller than this parameter (the "mesh size"). For any given mesh size you will have an uncountable infinity of possible partitions. The smaller the mesh, the finer-grained the partition and (hopefully), the more accurate the integral.

For any given partition, there is obviously a particular value for the sum of whatever you are integrating.

So can consider the set of sums for all partitions of a particular mesh size. Unless we are integrating an unbounded function or are integrating over an unbounded interval, this set will have both a least upper bound and a greatest lower bound. [Any non-empty set of real numbers that is bounded above has a least upper bound. That includes infinite sets].

We can take the limit of the least upper bounds as the mesh size goes to zero.
We can take the limit of the greatest lower bounds as the mesh size goes to zero.
If those match, we can use that shared limit as the result of the integration.

Now off I go to Google up a formal definition... Yup. Looks pretty close.

Lesbesgue integration is something else again.

 

[wrobel]

this one seems to be good for beginning
https://www.amazon.com/dp/B000KWQY9Y/?tag=pfamazon01-20
generations of Russian physicists and engineers came through it :)

 

[rudransh verma]

this one seems to be good for beginning
https://www.amazon.com/dp/B000KWQY9Y/?tag=pfamazon01-20
generations of Russian physicists and engineers came through it :)

Unavailable! Any pdfs?

 

[weirdoguy]

generations of Russian physicists and engineers came through it :)

Speaking of russian authors, polish version of this book by Зельдович was a great read when I was in high school:
https://www.amazon.com/dp/B000IW9YSO/?tag=pfamazon01-20
Unfortunately it's also unavailable.

Unavailable! Any pdfs?

Maybe you should check your schools library?

 

[Mark44]

“ To be more precise, it is the sum of the velocity at a certain time, let us say the ith time, multiplied by deltat.
$s=\sum_{i} v({t_i})\Delta t$”

Why don't you try this out with a specific example?
Here's one:
$v(t) = 9.8t$, for t in the interval [0, 5]. This represents the velocity in m/sec at time t seconds, of an object falling under the influence of gravity, assuming no air resistance.
For the first estimate, partition the interval into subintervals of length 1 second; i.e., $\Delta t = 1$.
Using the right endpoint of each subinterval, we have $v(1) = 9.8$, $v(2) = 19.6$, $v(3) = 29.4$, $v(4) = 39.2$, $v(5) = 49.0$.
With these figures, what do you get for $\sum_{i=1}^5 v({t_i})\Delta t$?

What do you get if $\Delta t = 0.5$? $\Delta t = 0.1$?

As $\Delta t$ gets smaller and smaller, is there some limiting value for the summation? This is what Feynman is getting at in the summation he wrote.

 

[Mister T]

$s= \lim_{\Delta t \rightarrow 0} \sum_{i}v({t_i})\Delta t$.
I don’t know how to take limit when summation is included. Help !

First you do the summation. Once you do that you take the limit.

 

[rudransh verma]

What do you get if Δt=0.5? Δt=0.1?

First I have to make the tables for 1, .5 and .1 sec. I didn’t make complete table for .1 sec, just up to 1sec. So maybe my answer may not be as accurate.
For $\Delta t= 1$ I got s=147 meters.
For $\Delta t= .5$ I got s= 134.75 meters
For $\Delta t= .1$ I got s= 5.39 meters
But I got the idea. Summation is going to a limiting value. When we take limit we will get the accurate value for s for up to 5 seconds.
This is called integration.

 

[Mark44]

For $\Delta t= .1$ I got s= 5.39 meters

The values for $\Delta = 1$ and for .5 look to be correct, but the one above is way off. The function $s(t) = \frac 1 2 at^2$ gives the value for s(5) as 122.5 meters.

But I got the idea. Summation is going to a limiting value. When we take limit we will get the accurate value for s for up to 5 seconds.

Yes, this is the idea behind the definite integral being the limit of a Riemann sum.

 

[A.T.]

First I have to make the tables for 1, .5 and .1 sec. I didn’t make complete table for .1 sec, just up to 1sec. So maybe my answer may not be as accurate.
For $\Delta t= 1$ I got s=147 meters.
For $\Delta t= .5$ I got s= 134.75 meters
For $\Delta t= .1$ I got s= 5.39 meters
But I got the idea. Summation is going to a limiting value. When we take limit we will get the accurate value for s for up to 5 seconds.
This is called integration.

See also: