riemann integration. introduction partitions norm of a partition
TRANSCRIPT
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?