Using the properties of the definite integral found in Theorem 5.2. So you'll see me using that notation in upcoming lessons.\(\require 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. When you're using the fundamental theorem of Calculus, you often want a place to put the anti-derivatives. 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. What are calculus's two main branches Calculus is divided into two main branches: differential calculus and integral calculus. It is concerned with the rates of changes in different quantities, as well as with the accumulation of these quantities over time. So this is going to equal capital f of 2 minus capital f of 0. Calculus is a branch of mathematics that deals with the study of change and motion. 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. The fundamental theorem of calculus states that differentiation and integration are inverse operations. Now it's also true that one third x cubed plus x plus 1 is an anti-derivative of x squared plus 1. 1 The Fundamental Theorem of Calculus Part 1
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