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. [SOUND]