1. Rieman sums
Lower sum
f (x) a h h h h h h h h h b

2. Rieman sums
Upper sum
f (x) a h h h h h h h h h b

3. Rieman sums
Rieman sum
f (x) a 1 2 3 4 5 6 7 8 9 b

4. Lower Rieman integral
If the limit for of the lower Rieman sum of a function f (x) exists, we call it the lower integral of f (x) over the interval [a,b].
where

5. Upper Rieman integral
If the limit for of the upper Rieman sum of a function f (x) exists, we call it the upper integral of f (x) over the interval [a,b].
where

6. Rieman integral
If, for a function f (x), both its lower and upper Rieman integrals exist and are the same, we say that f (x) is Rieman-integrable and call the common value of the lower and upper integral the Rieman integral of f (x). Formaly

7. Example
Find the lower and upper integrals of the function
over the interval [0,1].

9. How can we calculate ?
Put

12. Thus we have calculated that

13. Example
Calculate 
If follows from the symmetry of the figure that we can write

14. Partition into n subintervals each having a length of
Set up
To calculate first note that
with

15. Clearly,
is a geometric sequence, which we can sum up:

16. Let us now calculate the limit

17. Thus we have established that
This means that
and so

18. The previous examples have shown that finding the Rieman intergal of even relatively simple functions by using only the definition of the Rieman integral may be a very tedious job. Therefore we normally use other methods.

19. Theorem
If a function f (x) is integrable over the interval [a,b], then the limit of its Rieman sum is equal to the integral of f (x) over [a,b].
To prove this theorem, it is suficient to realize that
for every n, since
for any
the rest then follows from the squeezing rule.

20. Let an integrable function f (x) be defined on an interval [a,b] and let it have an antiderivative F (x), that is,
Let us divide [a,b] into n equal subintervals, denoting by xi the dividing points and putting a = x0, b = xn
a = x0 x1 x2 x3
xn-2 xn-1 xn = b
then we can write
where
using the mean value theorem.

21. Newton-Leibnitz formula
holds for all integers n and thus it could be easily proved that this equality will also hold if we take the limit for
Then
and since f (x) is
integrable over [a,b], we have
Newton-Leibnitz formula