
Integral
calculus 


The
definite integral 
Riemann
sum 
Calculating
a definite integral from the definition 
Calculating
a definite integral from the definition, examples 






Riemann
sum 
Until
now, in the definition of the sums, S
and s
we've used the maximum and the minimum values, M_{i}
and m_{i}
of a given continuous function
f, so that m_{i}
< f (x)
< M_{i} for x
in [x_{i}_{
}_{}_{
1}, x_{i}],
i = 1, 2 , . . . , n. 
Now,
if we arbitrarily choose a point x_{i}'
in every interval Dx_{i}
and make products f (x_{i}' )
Dx_{i},
then 
m_{i}
Dx_{i}
<
f (x_{i}' )
Dx
< M_{i}
Dx_{i},
i = 1, 2 , . . . , n 
and
by adding up 

The
lefthand and righthand sides of the above inequality are the sums, s
and S
respectively that, because of continuity of f,
tend to the same limit value I
when the number
of subintervals n
increases to infinity, such that the length of
every interval Dx_{i}
tends to zero, for any partition of the interval [a,
b]
and arbitrarily chosen
points x_{i}'
in the subintervals [x_{i}_{
}_{}_{
1}, x_{i}].
Hence,
the middle term 

called
a Rieman sum, will tend to the same limit value.

Therefore,
if f
is a positive continuous function on the interval [a,
b] then, the definite
integral of the function from a
to b
is defined to be the limit


Note
that the limit value of the sum changes as the number
of subintervals n
increases to infinity while the length
(Dx_{i})
of each tends to zero, for any partition of the interval [a,
b].


Calculating
a definite integral from the definition 
As the sequence of inscribed rectangles s
tends to the definite
integral increasingly while the sequence of circumscribed rectangles S
tends to the same value decreasingly then 

Thus,
we
can approximate the area under the graph of a function over the interval
[a,
b]
to any desired level of accuracy using
the Riemann sum of inscribed or circumscribed rectangles. 
The
area of the i^{th}
rectangle f (x_{i}' )
Dx_{i},
denoted as height times base, represents the i^{th}
term of Riemann sum
and is called the element of area. 
When
we use the partition of the interval [a,
b] into n
equal subintervals (regular partition) then 

be
the length of the intervals [x_{i}_{
}_{}_{
1}, x_{i}],
i = 1, 2 , . . . , n. 

Calculating
a definite integral from the definition, examples 
Example:
Evaluate 

where f (x)
=
1, using the
definition of the definite integral. 

Solution: Since
the graph of the constant f (x)
=
1
is the line passing through the point (0,
1)
parallel to the xaxis,
the region under the graph is the rectangle of
the base
=
(b

a)
with the height h
=
1. 
Thus,
the area 
A
=
(b

a) · 1
=
b

a, 
as
shows the right figure. Therefore, 





Example:
Evaluate 

where f (x)
=
x, using the
definition of the definite integral. 

Solution: Since
the graph of f (x)
=
x
is the line through the origin, coordinates of every its point y
=
x,
so
the region under the graph is the trapezium
with the height
b

a
and whose parallel sides are
a
and b. 
Let's
use the partition of the interval [a,
b] into n
equal subintervals,
so that Dx
=
(b

a) / n
and calculate 
the
lower sum s
of inscribed rectangles, as
shows the right 
figure.
If
we choose the point x_{i}'
to be the lefthand end of 
each
subinterval,
then 
x_{1}'
=
a
and
f (x_{1}'
)
=
a 
x_{2}'
=
a
+
Dx,
f (x_{2}'
)
=
a
+
Dx 
x_{3}'
=
a
+
2Dx,
f (x_{3}'
)
=
a
+
2Dx 
·
· ·
·
· · 
x_{n}'
=
a
+
(n

1)Dx,
f (x_{n}'
)
=
a
+
(n

1)Dx. 



We
use the Riemann sum to calculate the sum of inscribed rectangles with
bases of the same length, 

therefore, s
=
a
Dx
+
(a
+
Dx)Dx
+
(a
+
2Dx)Dx
+
·
· ·
+
[a
+
(n

1)Dx]Dx 
s
=
Dx[n
a +
Dx(1
+
2
+
·
· ·
+
(n

1))] 
to
calculate the sum of natural numbers inside of square brackets we use
the formula 
S_{n}
= [2a_{1} + (n 
1)d] for the sum of the
arithmetic sequence whose first term a_{1}
= 1 and difference d
= 1, 
so
we get the sum equals
(n

1)n
/ 2, and since Dx
=
(b

a) / n
then, 

Thus
the area under the graph of f (x)
=
x
over the interval [a,
b] 










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