So now we know what is and derivative.
Let us move on to the case of continuous summation.
And in order to illustrate this we're going to use most basic example,
which is the area under the curve.
So assume that we have some function f I've actually drawn a basic sine
wave here.
And some closed segment from A to B and we need to understand what
is the area under the curve is with one simple addition to it.
We need to think about oriented area of this figure under this big actually and
let me define this for you in the following manner.
Let's just come see the that the sine in before the area of the curve
is actually coincide with the sine of the function at this segment.
For example, if function lies are over 0 is a very big
positive wise and this positive value.
We get positive area and if it is,
if it coincides with negative values of the function, it is negative area.
Well, it's actually two places here.
So that's what's called a rated area here.
But how do we calculate one?
Well consider the most easy case for us to calculate area.
It's rectangle right, rectangular figures are extremely nice to calculate area.
But how are we supposed to some substitute this curve,
which is actually clearly not rectangular and by some rectangular alternative.
Well, if we draw something like this, this, this,
it's kind of rectangular but it's kind of not the same thing.
It's a false area, it's not not the same figure, right?
So we need to be more accurate and more specific, for example,
we need to use some more cena.
Much more cena rectangles and
thus we can get some understanding of the area of the curve by someone,
I mean this rectangles and around here.
So ideally what we should do, we should consider this rectangle has much,
much, more cena infinitesimally thinner.
And then we can get some understanding of area under this curve,
more exactly, right.
So let us consider all the necessary builds up for this constant.
Firstly, we should understand how do we find a layout for
this support for the bodice obviously rectangles.
In order to do so, we introduced the concept of partition of this segment.
Then we're going to call the partition as number of points of this segment.
For example M points here and we are going to order it from A to B.
As the for the sake of simplicity and
the maximal length of the segment in between neighboring dots.
We're going to call as a diameter of the partition in order to understand
how wide this partition in general is so if the diameter is rather small.
Thus, we know that since it's the maximal segments as well
there are segments are smaller than the diameter.
So we do not actually need to specify all the lengths here,
we just need to understand what the diameter is.
And I extremely should stress out to use that we are not working only
with uniform partitions which lies that all the lengths are the same.
No, there are beats really can be just chosen as you want,
you just need to clearly state what is the maximum value is.
So that's our partition, so
that's kind of borders of our rectangles, right?
We are going to build up something like this but if we now know the width of them,
we also need to specify the heights of them, right?
And another two so we are going to tag our partition and
well this partitioner normally known as stacked one in order to do so.
We are going to choose arbitrary some point, it's every segment here.
This, this, this and this, well, every segment has designated point.
We're going to call it here for each partition here and
thus, we get some understanding.
Because at this point we are going to call the heights of the respective
rectangular as the value and function or at this very point.
So if it is chosen something like this,
we're going to draw rectangular with this height.
So, that's our rectangular and
the area of this rectangular is going to be our language,
we are going to put an answer, right?
And remember that it's going to be a negative, so
we can come up with something which is called Riemann sum.
Which is basically the sum of all these areas with just defined, right?
So, what do we have we have partition sets view set, right?
Then at every segment of this partition with chosen some
point at which we actually consider a null function.
Such as we checked our partition, then we've calculated
the area of which rectangular for each segment of our partition.
By multiplying was value of the function at this partition by
widths of the segments of this partition.
Then by sum it needs we've got something which is somehow close to our rounds,
right?
If all the segments of our partition were extremely small,
extremely narrow that rectangular then we are going to have our area, right?
So the last thing we need to do is just take a limit, right?
Well, everybody knows when I'm speaking about extremely small,
I'm going to take a limit, so here it is.
So but I'm going to stress out the limit only consider
it via go into a half infinitesimal diameter of the partition.
And thus, we have infinitesimal each segment and let's kind of works for us.
A notation here is our Corman integral symbol, right?
And we here have subscript and
index here which describes our segment of integration.
So it's basically means that you need to consider
rented area under the curve of F at X from A to B.
That's what it is but the most complex scene here is
that we should consider this limit independently.
Independently from tagging,
when I say we choose our points arbitrarily in the tagging.
I mean that we are not actually in part about to choose special bands like
right and left and the maximal was a minimal one.
They just go on and go with any and we are do not specify an actual partition.
We just know is that each is going narrow, narrow and narrower, right?
So that's kind of complexity in the definition of our definite integral.
Sorry for this one, is kind of extremely complex,
thus is you may expect nobody actually uses it.
But we've, we are going to understand how to calculate
it by not the definition in up following videos.
So by now, we just do know what's the area under the curve is and
what is the definite integral is, so that's kind of nice.
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