So you'll see me using that notation in upcoming lessons. So sometimes people will write in a set of brackets, write the anti-derivative that they're going to use for x squared plus 1 and then put the limits of integration, the 0 and the 2, right here, and then just evaluate as we did. Together, the First and Second FTC enable us to formally see how differentiation and integration are. When you're using the fundamental theorem of Calculus, you often want a place to put the anti-derivatives. The Second Fundamental Theorem of Calculus is the formal, more general statement of the preceding fact: if f is a continuous function and c is any constant, then A(x) x cf(t)dt is the unique antiderivative of f that satisfies A(c) 0. This is the exact value for the area under that curve and we got it using just a couple of calculations, the anti-derivative evaluated at 2 minus the anti-derivative evaluated at 0. 2 is 6 thirds so this is 14 thirds or about 4 and 2 thirds. So this is going to be our be our answer. Now capital f of 2 is one third of 2 cubed, one third of 2 cubed plus 2 minus capital f of 0 one third of 0 cubed plus 0. So this is going to equal capital f of 2 minus capital f of 0. As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. So I need to evaluate this anti-derivative at 2 and then evaluate it at 0 and subtract. You can use any anti-derivative, it doesn't matter and that's why most people will choose to use the anti-derivative with a +0 here. Now it's also true that one third x cubed plus x plus 1 is an anti-derivative of x squared plus 1. I need an anti-derivative for it and an anti-derivative would be capital f of x equals one third x cubed plus x. In this integral from 0 to 2, this is my little f of x. So this is the integral I'm going to solve. That would be the integral from 0 to 2 of x squared plus 1 dx. 5) The term 'potential' was coined by George Green who lived from 1783-1841. Usually, to calculate a definite integral of a function, we will divide the area under the graph of that. So the exact area equals the definite integral of this function from 0 to 2. In one dimension, it reduces to the fundamental theorem of calculus R b a f0(x) dx f(b) f(a) 4) The theorem justi es the name conservative for gradient vector elds. The second fundamental theorem of calculus (FTC Part 2) says the value of a definite integral of a function is obtained by substituting the upper and lower bounds in the antiderivative of the function and subtracting the results in order. Says find the exact area under y=x squared plus 1 from x=0 to x=2. So let's see how that works out in an example. So you can evaluate a definite integral exactly using an anti-derivative and just evaluating it and subtracting. This b is the same as this b, this a is the same as this a. So again capital F is an anti-derivative of this inside function. If f is a continuous function and capital F is an anti-derivative of little f then the definite integral from a to b of little f of x dx is capital F of b minus capital F of a. I need to introduce a very important topic, "The Fundamental Theorem of Calculus." Here's the theorem right here.
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