1 / 21

RIEMANN INTEGRATION

RIEMANN INTEGRATION. INTRODUCTION. PARTITIONS. NORM OF A PARTITION. REFINEMENT OF A PARTITION. Upper and lower Riemann sums. EXAMPLES:. RIEMANN INTEGRAL. CONDITION OF INTEGRABILITY. Assignment. f(x)=x on [0,1] where P={0,1/3,2/3,1}?

xu
Download Presentation

RIEMANN INTEGRATION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. RIEMANNINTEGRATION

  2. INTRODUCTION

  3. PARTITIONS

  4. NORM OF A PARTITION

  5. REFINEMENTOF A PARTITION

  6. Upper and lower Riemann sums

  7. EXAMPLES:

  8. RIEMANN INTEGRAL

  9. CONDITION OF INTEGRABILITY

  10. 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?

  11. 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]

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

More Related