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AP.CALC:

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so I have a curve here that represents y is equal to f of X and there's a classic problem that mathematicians have long thought about how do we find the area under this curve maybe under the curve and above the x-axis and let's say between two boundaries let's say between X is equal to a and X is equal to B so let me draw these boundaries right over here that's our left boundary this is our right boundary and we want to think about this area right over here well without calculus you can actually get better and better approximations for it how would you do it well you could divide this section into a bunch of Delta X's that go from A to B they could be equal sections or not but let's just say for the sake of visualization so I'm going to draw roughly equal sections here so that's the first that's the second this is the third this is the fourth this is the fifth and then we have the sixth right over here and so each of these this is Delta X let's just call that Delta X 1 this is Delta X 2 this width right over here this is Delta X 3 all the way to Delta X n I'll try to be general here and so what we could do is let's try to sum up the area of the rectangles defined here and we could make the height maybe we make the height the based on the value of the function at the right bound it doesn't have to be it could be the value of the function someplace in this Delta X but that's one solution we're gonna go into a lot more depth into it in future videos and so we we do that and so now we have an approximation or we could say look the area of each of these rectangles are going to be f of X sub I or maybe X sub I is the right boundary the way I've drawn it times Delta X I that's each of these rectangles and then we can sum them up and that would give us an approximation for the area but as long as we use a finite number we might say well we can always get better by making our Delta X is smaller and then by having more debt more of these rectangles or get to a situation here we're going from I is equal to 1 to I is equal to n but what happens is Delta X gets thinner and thinner and thinner and we have an N gets larger and larger and larger as Delta X gets infinitesimally small and then as n approaches infinity and so you're probably sensing something then maybe we could think about the limit as we could say as n approaches infinity or the limit as Delta X becomes very very very very small and this notion of getting better and better approximations as we take the limit as n approaches infinity this is the core idea of integral calculus and it's called integral calculus because the central operation we use the summing up of an infinite number of infinitesimally thin things is one way to visualize it is the integral that this is going to be the integral in this case from A to B and we're gonna learn a lot more depth in this case it is a definite integral of f of X f of X DX but you can already see the parallels here you can view the integral sign as like a Sigma notation as a summation sign but instead of taking the sum of a discreet number of things you're taking an the sum of an infinitely an infinite number infinitely thin things instead of Delta X you now have DX infinitesimally small things and this is a notion of an integral so this right over here is an integral now what makes it interesting to calculus it is using this notion of a limit but what makes it even more powerful is it's connected to the notion of a derivative which is one of these beautiful things in mathematics as we will see in the fundamental theorem of calculus that integration the notion of an integral is closely tied closely to the notion of a derivative in fact the notion of an antiderivative in differential calculus we looked at the problem of hey if I have some function I can take its derivative and I can get the derivative of the function integral calculus we're going to be doing a lot of well what if we start with a derivative if can we figure out through integration can we figure out its antiderivative or the function whose derivative it is as we will see all of these are related the idea of the area under of curve the idea of a limit of something an infinite number of infinitely things than things and the notion of an antiderivative they all come together in our journey in integral calculus

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