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Integrate e-sign understanding

hello welcome to Tommy analytics it's going to be a video on integral notation we're going to start with explaining this part here inside the integral the f of X that refers to is the function that we're dealing with the function we're dealing with is x squared which is just a simple parabola and what we're concerned with is the area that lies underneath the function x squared between the points a and B and specifically I'm just going to roughly shade in all the area I'm going to take up all the blue all the yellow everything we approximate that area by using rectangles so I just pulled one of them out and if you remember the area is the width times the height now the width we refer to as the change in X and the height we refer to as f of X so here's the width change in X and the height we're going to use the left endpoint which maps to this point f of X which is just the Y value now the problem with using rectangles is that it does not account for the error that all this blue error underneath the curve so what we have to do is shrink the rectangles we need to shrink the width infinitesimally small so by that we're talking with that are so small it'll look something like this and it gets even smaller than that that's effectively what the change in X comes down to as we shrink the widths we can effectively fit in underneath the curve many many more rectangles so that will eliminate this blue error when we shrink the width infinitesimally small this change in X symbol becomes this symbol here DX now you want to add all of those rectangles up you want to sum them all so we can have a very good approximation to the area underneath the curve of x squared between a and B so the notation that we use for that is this symbol right here which is really just an S that is elongated that stands for summation to sum up all of the areas of the rectangles so we've explained what f of X means you've explained DX we've explained a and B and also this symbol right here now one other thing that you want to keep in mind is that you may see dy what that means is that the rectangles are along the y axis so the width would be the change in Y before we were referring to the change in X along X and shrinking the words down but over here the widths are with respect to Y so when you shrink this width down you get dy if you have any questions or comments please post below and thank you for watching