 In this video, we will understand how to figure out the area under a curved VT graph. So, for this case, let's say we want to figure out the area between the time limits T i and T f and we want to figure out all of this shaded blue area. We know that the area under a VT graph, that gives us the displacement of the object in motion. So, all of this blue shaded area, that gives us the displacement of this object that is moving, that is in motion. And we have already calculated areas under a VT graph. But in those cases, we calculated the area of a rectangle or a triangle and that gave us the displacement of the object. Now here, it is neither a rectangle or a triangle or a square in this one. So, how do we do it? How do we get the area of this shaded part? Well, we can get better and better approximations for it. And one way of doing that is, we can divide this area into sections, into sections of delta T's. And it does no matter if you make equal sections or sections of different width or of equal width. Let's just say, let's just say we divide this, we divide this into 10 sections of equal width. So, when we do that, this is how it can look like. And all of these widths, they are delta T. So, we can say that this is delta T1, this right here, this would be delta T2. This is delta T3, delta T4, 5, 6, 7, 8, 9, 10. So, 10 sections of same width, all, they have the same width delta T but we are just labeling it differently. And so, what we can do here is, we can try and sum up the areas of these rectangles. We know how to calculate the area of a rectangle, right? And we have 10 rectangles here. So, let's just find the area of these, the sum of the areas of these 10 rectangles and that will give us some approximation for the area, for the blue shaded area. So, if we do that, we know that the width for this one would be delta T1, delta T2, delta T3. So, if you look at the first rectangle, if you figure out the area of this one, we know that the width is delta T1 and this and the length, this would be, this could be, this could be V1 average, average V1. Because we are figuring out the velocity over a finite time interval, right? That is delta T1. And we know that average velocity, we know that average velocity, this was given by delta X divided by delta T. So, for a finite time interval, you get average velocity. For a finite delta T1, your average velocity is V1 average. And similarly for delta T2, it would be V2 average. Delta T3, V3 average so on and so forth till delta T10. So, now we know the area of this, let's say this red shaded rectangle that is V1 average into delta T1. And we can add all of these rectangles. So, when we do that, this is how, this is how it could look like. V1 average into delta T1 plus till V10 average into delta T10. We can write it more like in a compact manner. This is a very long way of writing it. So, let's do that. We can give a general I to show 1, 2, 3. So, V I average into delta T I. Here I could be 1, 2, 3, 4. And to show that we can add a sigma sign. So, I starts from 1 and goes till 10 and we are adding all of that. This is the sum of the areas of all of these rectangles. Now, what we can do is we can get better and better approximations of the area that we are calculating. Because we can see that the area of the rectangles in this case, it's missing out on some parts. It's missing out this region, missing out this region, this region. This region is not being included in the total area. So, one way to deal with it is to increase number of rectangles or the sections that you are dividing the blue shaded area in. So, if we do that, let's say if we make 20, if we divide it in 20 sections, if we make 20 rectangles, something like, let's say we divide this, then we divide this. So, we'll have to rub off, we divide this. So, basically we are now dividing it into 20 sections. Let me show you how a couple of rectangles would look like so that you get some idea. So, if we are making 20 such rectangles, this is how thin they could look like. And now we are adding 20 of such rectangles. So, we can change this to 20. And now you're getting a better approximation of the area because you see these smaller triangles, now you're taking care of some of them, not entirely. The approximation is somewhat better for the area, not perfect, but still somewhat better. If you make 30 or 40 or 50 such sections of the shaded area, then you will keep on getting better and better approximations. And if you kept on making this interval of delta t smaller and smaller, if you keep on adding more and more sections, you're just getting better and better approximations. So, let's first make it more general. Let's say we have n, we have n number of such sections, such rectangles. And if you keep on decreasing delta t, if you keep on making it thinner and thinner, so that you get an infinite number of sections in this rectangle, that happens when n, that will happen when n approaches infinity, so that is how we can write it. The variable n is approaching infinity, delta t is infinitely thin, and you have infinite number of sections in this blue shaded area. So, that can give you a great approximation, because it's much better than 10, 20, 30, it's now you have infinite number of sections or rectangles. This is your displacement, the blue shaded area. And now since delta t has approached zero, delta t has almost approached zero, this is now almost zero, we can move from the notion of average velocity and towards instantaneous velocity, because that's what instantaneous velocity was, right? Instantaneous velocity is the velocity at one particular time instant, that happens when delta t is almost zero. So, we can remove the average part from over here, and when we do that, this is all that remains, the instantaneous velocity into delta ti. And this notion of getting better and better approximations as n approaches infinity, this is the core, this is the core basic idea of integral calculus. And we call it integral because the operator that we use to represent this, it looks like this, this is called an integral. This right here, this right here tells you the area under the vt graph from the limit ti to tf. And here v is really a function of time, this could be any function of time, it could be 2t square plus 3, 5tq plus 4, it could be really anything, it's just a function of time. So, whenever we do come across this integral of vdt from some time limits, it basically means the area under the vt curve from the limit ti till the limit tf.