MAT125B Lecture Notes

1 Riemann integration 2 1.1 Partitions and Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 A criterion for integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Upper and Lower Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The refinement of a partition . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Properties of upper and lower sums . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Definition 1.4 implies Definition 1.11 . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Monotonic and piecewise continuous functions are integrable . . . . . . . . 10 1.8 The Riemann integral is linear . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Further properties of the Riemann integral . . . . . . . . . . . . . . . . . . . 13 1.10 Interchanging limits with integrals . . . . . . . . . . . . . . . . . . . . . . . 15 1.11 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . . 16 1.12 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27