Definite Integral
The definite integral of a function over an interval is a number that
expresses the total change in any antiderivative of the function over this
interval.
Definite Integral
Dr.Horváth Gézáné Ph.D.
Lecture 5

Example
If
f(x) = 2x + 1
F(x) = x2 + x + C any antiderivative of f(x).
The total change in F over interval [1, 3] is thus
F(3) – F(1) = 9 + 3 + C – ( 1 + 1 + C ) = 10.
The constant of integration is subtracted out.
Definition 1
If f is a function defined on the closed interval [a, b], then
the definite integral of f over [a, b], symbolised by b
f ( x)dx
a
,
is the total change in any antiderivative of f(x) over the interval.
Dr. Horváth Gézáné PhD
Lecture 5
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Definite Integral

2
Riemann Integral
Definition 2
(Newton – Leibniz formula)
If f is a function with an antidervative F over the interval [a, b], then
the definite integral of f from x = a to x = b is



b
f ( x)dx
F ( x)
b
a
F (b) F (a)

a
The numbers a and b are referred to as the lower and upper limits
of integration. This is the Newton – Leibniz formula.


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Definition 3
If f is continuous over the closed interval [a,b] and F is an
antiderivative of f, then
n

F (b) F (a)
n
Let f be a continuous and non-negative function defined on the closed
interval [a, b].
From each subinterval we can choose a single number i .
Find the functional value f( i ) and form the sum
f( 1) x + f(
b
f ( x)dx
Let n be any positive integer number.
Suppose the closed interval [a, b] is divided into n equal subintervals.
The end points of the sub intervals are
a = x0 < x1 < x2 < … < xn =b
1
b a
x
(b a)
In each of subintervals will have length x, where
2
) x + f(
3
) x +…+ f(
n
) x
This sum is Riemann sum for f over [a, b].
a
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Lecture 5
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Riemann Integral
Riemann Integral
If we construct a rectangle of height f( i ) on each subinterval [xi-1 , xi ], the
area of rectangle will be f( i ) x .
The total area of the rectangles approaches the area under graph of f
from a to b, which is the definite integral of f from x = a to x = b.
This Riemann sum is the sum of areas of all such rectangles.
f( 1) x + f(
2 ) x + f(
3 ) x +…+ f(
n) x
Therefore the limit of Riemann sums for f over [a, b] as x
definite integral.
0 is the
The Definite Integral as the Limit of Riemann Sums is the following
b
f ( x)dx
lim f ( 1 )
x
As n
, x
0, in other words, as n becomes lager and lager, we can
subdivide the interval [a, b] into more and more subintervals of shorter and
shorter length.
The limit of Riemann sums is the Definite Integral.
Lecture 5
f ( 2 ) ...
0
f ( n ) x.
a
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Dr. Horváth Gézáné PhD
Lecture 5
5
Theorems of Integration
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Properties of definite integrals
If the function f(x) can be integrated on the closed interval [a, b] then
f(x) is bounded on [a, b].

Property 1
a
f ( x)dx
0
a
If the function f(x) is continuous on the closed interval [a, b] then f(x)
can be integrated on [a, b].

Property 2
If c is any number such that a < c < b, then
b

If the function f(x) is defined and monotone on the closed interval
[a, b] then f(x) can be integrated on [a, b].


Property 3
Let a function f(x) be bounded on a closed interval [a, b] and let f(x)
have a finite number of discontinuities, then f(x) can be integrated on
[a, b].
Lecture 5
c
f ( x)dx
a
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f ( x)dx
c
For all a and b
b
a
f ( x)dx
a
Lecture 5
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b
f ( x)dx
a
f ( x)dx
b
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Application of Definite Integral
Application of Definite Integral
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
The average value of a function f over the closed [a, b] is
M


1
b
b aa
The marginal cost function for producing q unit of a certain product
is
C’(q) = 5 – 0.01 q
Find the total cost incurred by increasing the production level from
200 to 400 units.
b
f ( x)dx
(b a) M
f ( x)dx
a
M is the value of f(x) over [a, b] such that the rectangle with base
[a, b] and height M has the same area as the region under the
graph of f from x = a to x = b.

Example 3
A company has determined that its profits are increasing at a rate
given by the following marginal profit function
’(q) = 100 + 200q– 12q2
where q represents the number of units sold.
Determine the total profit which would be generated by increasing
the number of unit sold from 10 to 16.
Example 1
Over the course of each month (30 days) a retailer’s inventory
declines linearly from 300 units to 60 units. Find the retailer’s
average daily inventory for the months?
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Lecture 5
Example 2
Lecture 5
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Application of Definite Integral
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Example 4
If p(x) represent the demand function for a certain product, then the
definite integral represent the total revenue generated by selling q
q
units of the product.
p( x)dx
0
For each the following demand functions, determine the total revenue
generated by selling 100 units.

(a)
p(x) = 50 – x/8

(b)
p(x) = 900/(x-10)2 + 20
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