1. RIEMANN INTEGRATION

2. INTRODUCTION

3. PARTITIONS

4. NORM OF A PARTITION

5. REFINEMENT OF A PARTITION

6. Upper and lower Riemann sums

7. EXAMPLES:

8. THEORMS OF UPPER AND LOWER SUM

11. RIEMANN INTEGRAL

17. CONDITION OF INTEGRABILITY

20. Assignment f(x)=x on [0,1] where P={0,1/3,2/3,1}? If P is a partition of interval [a,b] and f is a bounded function defined on [a,b], then L(f,P)  U(f,P)? f(x)=sinx on [0,  /2] where P={0,  /6,  /3,  /2}? State and prove Darboux theorem? State and prove necessary and sufficient condition of integrability? Every monotonic and bounded function is integrable?

21. A continuous function on a close interval is integrable on that interval? Show that greatest integer function f(x)=[x] is integrable on [0,4] and  [x]dx=6? let f be a bounded function such that the set of points of discontinuity of f on [a,b] then f is integrable on [a,b]? show that f(x)=|x| is integrable on [-1,1]

22. TEST Attempt any three: State and prove Darboux theorem? Evaluate  x m dx,m≠-1 on [a,b]? Prove the condition of integrability? Give an example of a function which is bounded but not integrable?