And how would we represent that? How would we denote that? Well, we could denote thatĪs the definite integral between a and b of f of t dt. To this magenta area, what are you left with? Well, you're left with this So if you have this blueĪrea, which is all of this, and you subtract out It's all of thisīusiness right over here. Literally the area between c and a under theĬurve lowercase f of t. X- it's going to be equal to the definite integralīetween c and b of f of t dt, which is justį of a, which is just the integral betweenĬ and lowercase a of lowercase f of t dt. We just literally replace the b where you see the So F of b- and we're going toĪssume that b is larger than a. So let's think about what F ofī minus F of a is, what this is, where both b and a areĪlso in this interval. To use to actually evaluate definite integrals. Second fundamental theorem of calculus, which we tend Now, what I wantįundamental theorem of calculus to the second part, or the Capital F of x is differentiableĪt every possible x between c and d, and theĭerivative of capital F of x is going to be equal Over this interval, then F of x is differentiable atĮvery x in the interval, and the derivative of capitalį of x- and let me be clear. Theorem of calculus tells us that if f is continuous So this right over here,į of x, is that area. Where f is continuous, under the curve- so it's theĪrea under the curve between c and x- so if this is x right And the reason why I'm usingĬ and d instead of a and b is so I can use aĭefined as the area under the curve betweenĬ and some value x, where x is in this interval Got some function f that is continuous over Simple question, how do we denote the area under the curve of f(t) from a to b? Why, its just the definite integral of f(t) from a to b!Īnd since we know the relationship between f(t), f(x), and F(x), we can easily evaluate the definite integral of f(t) from a to b! So the same applies here, when you subtract the smaller quantity F(a)= from the larger quantity F(b), you get the quantity of the area between a AND b. Let me ask a question, what happens when you subtract a smaller number from a larger number? You get the distance between the numbers! In other words, if you take the anti-derivative of f(x), you get F(x), which shows us that if you have f(x), you can find F(x). The first part of the fundamental theorem of calculus tells us that the derivative of F(x) (which is just the rate of change of the area under f ) is equal to the function f(x) (which is exactly the same function as f(t) just with a different variable). That area is going to be larger than the area from c to a which equals F(a). So, you start with the area under the curve of the function f(t) from c to b which equals F(b). For anyone who is confused, try thinking of it this way.į(x) is the area under the curve of f(t) from some value c to any input value of your choice, x.
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