GROUP 3 RIEMANN NTEGRATION ON R^n

1 Riemann Integration 2

1.1 Partitions and Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Definition  (Partition P of size > 0) . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Definition (Selection of evaluations points zi) . . . . . . . . . . . . . . . . 2

1.1.3 Definition (Riemann sum for the function f(x)) . . . . . . . . . . . . . . 2

1.1.4 Definition (Integrability of the function f(x)) . . . . . . . . . . . . . . . 2

1.1.5 Definition (Notation for integrable functions) . . . . . . . . . . . . . . . . 3

1.2 Upper and Lower Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Definition (Mi and mi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Definition (Upper and Lower Riemann Sums) . . . . . . . . . . . . . . . 3

1.2.3 Definition (Integrability of f(x) in terms of L(f) and U(f)) . . . . . . . . 3

1.2.4 Example

1.2.5 Definition (Refinement of Partitions) . . . . . . . . . . . . . . . . . . . . 4

1.3 Properties of Upper and Lower Sums . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Theorem (Cauchy Criterion for Integrability in Terms of Upper and Lower

Sums) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The Riemann Integral is Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Further Properties of the Riemann Integral . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 Theorem (Fundamental Theorem of Calculus) . . . . . . . . . . . . . . . 5

2 Preliminaries 6

2.1 Definition(An interval) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Definition(Length of Interval) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Definition(- neighborhood of an Interval) . . . . . . . . . . . . . . . . . . . . . 6

2.5 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.1 Definition(n-cell) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 The Riemann Integral In n-Variables 7

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Upper and Lower Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.2 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Properties of Riemann Integral in n Variables . . . . . . . . . . . . . . . . . . . 9

3.4 Iterated Integrals and Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.2 Corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Note

Overall Rating

0

5 Star
(0)
4 Star
(0)
3 Star
(0)
2 Star
(0)
1 Star
(0)
APA

Ayadi, F. (2018). GROUP 3 RIEMANN NTEGRATION ON R^n. Afribary. Retrieved from https://track.afribary.com/works/group-3-riemann-ntegration-on-r-n

MLA 8th

Ayadi, Fayowole "GROUP 3 RIEMANN NTEGRATION ON R^n" Afribary. Afribary, 29 Oct. 2018, https://track.afribary.com/works/group-3-riemann-ntegration-on-r-n. Accessed 25 Dec. 2024.

MLA7

Ayadi, Fayowole . "GROUP 3 RIEMANN NTEGRATION ON R^n". Afribary, Afribary, 29 Oct. 2018. Web. 25 Dec. 2024. < https://track.afribary.com/works/group-3-riemann-ntegration-on-r-n >.

Chicago

Ayadi, Fayowole . "GROUP 3 RIEMANN NTEGRATION ON R^n" Afribary (2018). Accessed December 25, 2024. https://track.afribary.com/works/group-3-riemann-ntegration-on-r-n