The Fundamental Theorem of Calculus
DOWNLOAD
FREE
|
The fundamental theorem of calculus
Equations are omitted for technical reasons - download the original pdf
The fundamental theorem of calculus states that integration is the inverse operation of differentiation. Definition, primitive Let f be integrable in the closed interval [Equation goes here - download the original to see it.]. Define the function F by [Equation] F is called the primitive of f. Theorem The primitive F of a function f integrable in the closed interval [Equation] is continuous. Proof [Equation] Let [Equation] on the interval, [Equation]. Then [Equation] and by result (5) from the preceding section [Equation] Hence [Equation] Hence [Equation] So F is continuous The fundamental theorem of calculus Let f be integrable in the closed interval [Equation] and let F be its primitive. Then [Equation] Proof Let [Equation] on the interval, [Equation]. As in the preceding proof [Equation] Hence [Equation] Hence [Equation] Now [Equation] and [Equation] Hence Equation]
|
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
|
