RIEMANN INTEGRATION

INTRODUCTION

PARTITIONSPARTITIONS

NORM OF A PARTITION

REFINEMENT OF A PARTITION

Upper and lower Riemann sums

EXAMPLES:

RIEMANN INTEGRAL

CONDITION OF INTEGRABILITY

Assignmentf(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?

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]

TEST

Attempt any three:

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?