The Fundamental Theorem of Calculus
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Integration as the Sum of Approximations
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We are required to find an approximation to the area under a given curve, represented by the function [Equation goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] We will approximate the area by rectangles. Each rectangle will have the same width. The width is denoted by [Equation] which means “small increase in x”. [Diagram goes here - download the original to see it.] As the rectangles get smaller and smaller – that is, as the width of the rectangle gets smaller – the sum of the area of the rectangles gets closer and closer to the area under the graph. [Equation] [Equation] We will introduce the symbol [Equation]to mean “sum”, and write this as [Equation]. In the limit as [Equation] gets smaller and small and approaches zero this area becomes a better and better approximation to the exact area under the curve. This limit is denoted by [Equation], which we introduced right at the start of this chapter as the expression for the exact area under the curve [Equation]. [Equation]The symbol dx indicates that in the expression [Equation] the limit [Equation] has been taken so that the area has become exact
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Contents of
The Fundamental Theorem of Calculus
1 Text: The fundamental theorem of calculus
2 Fundamental Theorem of Calculus
3 Infinite series
4 Integration as the reverse of differentiation
5 Convergence and divergence of series
6 Integration as the Sum of Approximations
7 The dissection
8 Properties of infinite series
9 Criteria for convergence
10 Definition, dissection
11 Definition, upper and lower approximative sums
12 Definition, upper and lower integrals
13 Definition, integrable function
14 Theorem, continuous functions are integrable
15 Properties of the integral
16 The fundamental theorem of calculus
17 Theorem, evaluation of the definite integral
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