Presntation for the post of lecturer in Mathematics

Comparison of Riemann Integration and Lebesgue Integration Theory. by KIFAYAT ULLAHDemonstration for the post of Lecturer in Mathematics,University of Science & Technology, Bannu

R.I Theory• First of all it is mentioned that both the theories is used for

finding the area under the curve of a function.

• Let be any bounded function defined on closed interval [a,b]. Let M and m be the lub and glb of f(x) in [a,b].

Define , Then clearly is an area very greater then the required area and is the area very small then the required area.

Let us divide [a,b] into two subinterval namely [a,c] and [c,b]. And Let be the upper, lower sum in the interval [a,c] and [c,b] respectively.

R.I TheoryAgain define , these are upper and lower sum

for this partition.

Note that , but still upper sum is greater then and lower sum is less then the required area. It means that the upper sum decreases and lower sum increases and converging toward the required area.

Let us devide[a,b] into n subintervals and let the points of partition are

R.I TheorySome diagrames about upper sum and lower

sumUpper sum

Lower Sum

R.I TheoryThen the upper and lower sum for n

subintervals is ,Where are the lub’s and glb’s in the respective subinterval .

Take if exists.

If I=J then the function is called R.Integrable, and the common value is the required area denoted by

.

L.I TheoryFor lebesgue integration theory first we have to

know about the measure of a set which generalized the concept of length of an interval.

The second difference in the technique is that, here we take partition of the rang instead of the domain.

First for bounded FunctionsLet f(x) be a bounded function defined on a set

E such that , then the first upper and lower sum is defined as ,

which plays role of upper and lower sum as in R.I theory.

L.I Theory Continuing in the same way let us divide into

n subintervals and let the points of partition are , and define

Then the upper and lower sum becomes ,

Take

L.I Theory If I=J then the function is called L.Integrable, and

the common value is the required area denoted by

Now for Unbounded FunctionsNow let f(x) is unbounded function, such that .

Then define .Clearly is bounded s.t so that

L.I Theory If is unbounded function, then , so the area

under the curve of can found by above case, and then multiply answer with -1 to get area under the curve of .

If unbounded have an arbitrary sign then define

& Clearly , so .

Comparison R.I theory work only for bounded function while L.I theory

works for bounded as well as for unbounded function.

R.I theory only work for the function defined on a compact interval while L.I theory work for function defined on interval as well as on set.

L.I theory is the generalization of R.I theory, every proper R.Integrable function is L.Integrable but the converse is not true.

R.I theory take partition on x-exis while L.I theory use y-exis for partition.

Every continuous function is R.Integrable as well as L.Integrable.

Example Here we have an example of function f(x)

which is L.Integrable but not R.Integrable. Where

The discontinuous function may or may not be R.Integrable as well as L.Integrable. Like following function is R.Integrable:

The unbounded function which is L.Integrable: