The Fundamental Theorem of Calculus
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The dissection
Equations are omitted for technical reasons - download the original pdf
In the following diagram [Diagram goes here - download the original to see it.]
the area under the curve is approximated by a sum of the areas of rectangles. It is said to be an approximation of the area from below. It is also possible to approximate the area from above, as the following diagram illustrates. [Diagram goes here - download the original to see it.] So the area lies between the two approximations, one from above and the other from below. In the limit, as the the width of the rectangles [Equation] both approximations converge on the same value, which is the actual area.
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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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