What will be the amount of water accumulated in an hour?
One hard way would be to count the number of drops and calculate.
Is there a better way?
Let's say we analyze the situation and get that
the rate of DROPS FALLING PER MINUTE  is constant.
It is 100 drops per minute.
Then we can easily find the answer.
In an hour 6000 drops will be accumulated.
But what if the rate is not constant or if the water is flowing continuously?
Let's consider another situation.
We want to find the number of vehicles traveling on this road in a day.
On a busy highway, we can think of the stream of vehicles
as a CONTINUOUS  quantity like water.
So we see that the problem could arise when we have to
add a continuous quantity or even a discrete one.
Addition of numbers is one of the most basic things
that we learn in our childhood.
We get more efficient and faster at it once we learned
how to multiply numbers.
But when addition becomes continuous,
we have to upgrade this skill.
In this video and a few upcoming ones,
we will take our skill of addition to the next level.
The upgrade is called the process of
integration.
Do you remember that in one of our previous videos,
we saw that integration is used to find the area or the volume of different shapes?
So, how can it be used to solve this problem?
Actually, these two things are connected to each other.
There is a relation between geometry and addition.
For example, we can easily count the number of points currently on the screen.
But what if I ask you to count now?
You could say that this question is absurd.
We cannot count the number of points of a line as its continuous.
For any two points on it, we can always find a point between them.
So instead of total number of dots we define the length of a line.
This was just a glimpse of the idea.
Let's see what exactly do we mean by this.
I want you to imagine a situation!
Recently, you are hooked to a video game.
As you keeps winning,
you are rewarded with points and you just want to keep collecting them.
You wonder how many points you would collect after three weeks.
Luckily the game has a feature
that shows the rate at which you are earning the points.
It shows that after "t" weeks your rate is "t" squared thousand points per week.
So assuming that you keep earning points at this rate,
how many points will you have after three weeks?
You could probably say it's easy.
From this relation we can find the rate for each week. In week one,
the rate will be thousand points per week. In week two,
four thousand points per week and in week three it will be nine thousand.
So according to these rates, in the first week
the points earned will be a thousand,
in the second week four thousand and in the third week, nine thousand.
So fourteen thousand points in total.
But is this correct ?
Actually not. In these calculations,
we directly assumed that the rate is constant during each week.
But it's not constant. This function implies that the rate is continuously growing.
Its graph looks like this.
So how do we find the total number of points?
For this, we need to understand what does this calculation tell us graphically.
Let's continue this in the next part.
Look at this graph.
Here, we have the number of weeks on the x-axis,
and the rate of earning points on the y-axis.
This calculation implies that the rate is constant during each week.
So it can be represented by the horizontal lines like this.
Now notice one interesting thing.
Look at these rectangles.
The area of the first rectangle
will be equal to the total number of points earned in the first week.
How?
The height of this rectangle is equal to the rate of one thousand points per week,
and its width is equal to one week.
So its area will be equal to one multiplied by one thousand
which equals one thousand points.
So we see that the area below these horizontal lines, tells us
the total number of points earned during that duration.
Can you find the areas of the other two rectangles?
For the second rectangle,
it will be equal to four thousand points and
for the third it will be nine thousand points.
So we get the sum of the areas of these three rectangles,
to be equal to fourteen thousand points.
This is the answer we got using this calculation.
So we see that the process of addition here, simply
becomes finding the total area covered by these rectangles.
But now we know that the rate is not constant during each week.
Notice in the graph that in each week,
the actual rate is lower than the rate we considered here.
Consider the first half of week one.
Here, the rate will not be equal to thousand points per week.
At the end of the first half of week one,
the rate will be one-fourth of a thousand that is
two hundred and  fifty points per week.
So let's consider this as the constant rate throughout
the first half of week one instead of one thousand.
This will give us a better estimate of total number of points.
Can you tell me the total points earned in this duration?
Yes, it will be equal to the area of this rectangle.
125 points!
If we do this for each half of all the three weeks, we will get this graph.
Now we know that the total points earned will be equal to
the sum of the areas of these rectangles.
After doing these calculations and adding, we will get the answer
as eleven thousand three hundred and seventy five points.
Earlier, we got the total as fourteen thousand points, and now it's this number.
This is a better estimate than the previous one
because the rate considered here is closer to the graph.
So, what should we do next to improve our estimate further?
Correct!
If we the consider an even shorter duration of a week.
For example a day, we will get an even better estimate.
As we keep decreasing the width of the rectangles further,
we see that this total area will approach the area under this curve.
So we see that the problem of finding the total number of reward points,
becomes the problem of finding the AREA UNDER THE GRAPH of this function.
This is the connection between geometry and addition we mentioned earlier.
In abstract terms we can say that,
if we have a function which represents the rate of some quantity,
then the area under the graph of that function
tells us the total amount of that quantity.
Now we need to know, how to find the area under the graph of a function.
Integration is the process used to find this area.
And this area is called the integral of the function.
In the next video, we will see how to find the integral of a function.
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