The Fundamental Theorem of Calculus
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Definition, upper and lower approximative sums
Equations are omitted for technical reasons - download the original pdf
Let [Equation] on the rth interval, [Equation]. Define the upper sum as the sum of the areas of n rectangles with base [Equation] and height [Equation]. Define the lower sum similarly; that is [Equation] The actual area, R contained between the lines [Equation] under the curve[Equation] is contained between these two sums; that is [Equation] Let [Equation] stand for any value of x in the rth subinterval; that is [Equation] Let [Equation] Then [Equation] We need to show that in the limit as the number of subdivisions in the dissection [Equation] then [Equation]. But this is complicated because the value of [Equation] depends on both [Equation]. However, one way to circumvent this difficulty is to use upper and lower bounds.
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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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