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let's say I have some function f that is continuous continuous on an interval continuous on an interval between a and B and nine of these brackets here so it also includes a and B in the interval so let me graph this just so we get a sense of what I'm talking about so that's my vertical axis this is my horizontal axis I'm going to label my horizontal axis T so that we can save X for later I can still make this Y right over there and let me graph this right over here is the graph of y is equal to f of T now our lower endpoint is a so that's a right over there our upper boundary is B upper boundary is B let make that clear and actually just to show that we're including that endpoint let me make them bold lines filled in lines so lower boundary a upper boundary B we're just saying and I've drawn it this way that F is continuous on that now let's define some new function let's define some new function that's the area under the curve between a and some some point that's in our interval let me pick this right over here X so let's define some new function to capture to capture the area under the curve between a and X well how do we how do we denote the area under the curve between two endpoints well we just use our definite integral that's our Riemann integral it's really that right now before we come up with the conclusion of this video it really just represents the area under the curve between two endpoints so this right over here we can say is the definite integral the definite integral from A to B from a to X I should say from a to X of F of T of F of T DT D DT now this right over here is going to be a function of X and let me make it clear where X where X is in the interval between a be this thing right over here is going to be another function of X this value is going to depend on what X we actually choose so let's define this as a function of X so I'm going to say that this is equal to upper case upper case f of X so all fair and good upper case f of X is a function if you give me an x value that's between a and b it'll tell you the area under lower case F of T between a and X now the cool part the fundamental theorem of calculus the fundamental theorem of calculus tells us let me write this down because this is a big deal fundamental theorem theorem that's not an abbreviation theorem of calculus tells us that if we were to take the derivative of our capital F so the derivative let me make sure I have enough space here so if I were to take the derivative of capital F with respect to X which is the same thing as taking the derivative of this with respect to X which is equal to the derivative of all of this business let me copy this so copy and then paste which is the same thing I've defined capital F as this stuff so if I'm taking the derivative of the left hand side that's the same thing as taking the derivative of the right hand side the fundamental theorem of calculus tells us that this is going to be equal to it's going to be equal to F lowercase F of X now why is this a big deal why does it get such an important title as the fundamental theorem the fundamental theorem of calculus well it tells us that for any continuous function f if I define a function that is the area under the curve between a and X right over here that the derivative of that function is going to be F so let me make it clear every continuous function every continuous F has an antiderivative has an antiderivative anti derivative derivative capital F of X that by itself is a cool thing but the other really cool thing or I guess these are somewhat related remember coming into this all we did we just viewed the definite integral as symbolizing as the area under the curve between two points that's where that riemann definition of integration comes from but now we see a connection between that and derivatives when you're taking the definite integral one way of thinking especially if you're doing a definite integral between a lower boundary and an x one way to think about is is you're essentially taking an antiderivative so we now see a connection connection and this was why it is the fundamental theorem of calculus it connects differential calculus and integral calculus connection between between derivatives derivatives and it or maybe I should say anti derivatives derivatives and integration which before this video we disputed integration as area under curve now that we see it has a connection to derivatives well how would you actually use the fundamental theorem of calculus so maybe in the context of a calculus class and we'll do the intuition for why this happens or why this is true and maybe a proof in later videos but how would you actually apply this right over here well let's say someone told you so let's say that someone told you that they want to find the derivative let me just in a new color just to show this is an example let's say someone wanted to find the derivative with respect to X of the integral from I don't know I'll pick some random number here so pi to X of I'll put something crazy here cosine squared of T over over the natural log of t minus the square root of T DT so they want you to take the derivative with respect to X of this crazy thing remember this thing if we this thing in the parentheses is a function of X its value it's going to have a value that is dependent on X you give it a different X it's going to have a different value so what's the derivative of this with respect to X well the fundamental theorem of calculus tells us it can be very simple we essentially we essentially and you can even pattern match up here and we'll get more intuition of why this is true and view videos but essentially everywhere where you see this right over here is an F of T everywhere you see a T replace it with an accident becomes an F of X so this is going to be equal to cosine squared of x over the natural log of X minus the square root of x you take the derivative of the indefinite integral where the upper boundary is X right over here it just becomes whatever you were taking the integral of that as a function instead of T that is now a function of X so it can really simplify sometimes taking a derivative and sometimes you'll see on exams these trick problems where you have this really hairy thing that you take a definite integral of and then take the derivative you just have to remember the fundamental theorem of calculus the thing that ties it all together connects derivatives and integration that you can just simplify it by realizing that this is just going to be instead of a function f of lowercase F of T it's going to be lower case f of X let me make it clear in this example right over here this right over here was lower case F of T and now it became lower case f of X this right over here was our a and notice it doesn't matter what the lower boundary of a actually is you don't have anything on the right hand side that is in some way dependent on a anyway hope you enjoyed that in the next few videos we'll think about the intuition and do more examples making use of the fundamental theorem of calculus

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