ميةس محمد بن سعود اامممعة ا جاداريةعلوم اد والقتصا كلية استثمارتمويل وا قسم الAl-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences Department of Finance and Investment Financial Mathematics Course FIN 118 Unit course 4 Number Unit Integral Calculus Unit Subject Dr. Lotfi Ben Jedidia Dr. Imed Medhioub 1
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
جامعة اإلمام محمد بن سعود اإلسالمية
كلية االقتصاد والعلوم اإلدارية
قسم التمويل واالستثمار
Al-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences
Department of Finance and Investment
Financial Mathematics Course
FIN 118 Unit course
4 Number Unit
Integral Calculus Unit Subject
Dr. Lotfi Ben Jedidia Dr. Imed Medhioub
1
1. Integral calculus : Definition
2. Indefinite integral
3. Definite integral
4. Some rules of integral
5. Area between two curves
2
We will see in this unit
Learning Outcomes
3
At the end of this chapter, you should be able to:
1.Understand what is meant by “integral of function”.
2. Find definite or indefinite integrals.
3. Calculate the Area Between Two Curves.
Integral calculus
Frequently, we know the rate of change of a
function and wish to find the original
function . Reversing the process of
differentiation and finding the original function
from the derivative is called integration or anti-
differentiation. The original function, , is
called the integral or antiderivative of .
Thus, we have
4
xf '
xf
xf
xf '
cxfdxxf '
Example 1:
1/ Find the derivative of , ,
2/ Find the antiderivative of the results of
question 1.
Solution:
1/ , ,
2/ ,
5
cxf 1
0'1 xf
cdxdxxf 0'1
xxf 2 23 xxf
1'2 xf xxf 2'
3
cxdxdxxf 1'2
cxdxxdxxf 2'
3 2
Integral calculus
Indefinite Integral
• The indefinite integral of a function is a
function defined as :
• Every antiderivative F of f must be of the form
F(x) = G(x) + c, where c is a constant (constant of
integration)
!!!
6
cxFdxxf
cxdxx 2 2
Represents every possible antiderivative of 2x.
Definite integral
If f is a continuous function, the definite
integral of f from a to b is defined as:
7
aFbFdxxfb
a
n
k
k
b
a nxxfdxxf
1
)(lim)(
kk xxn
abx
1
An integral = Area under a curve
Integral Calculus
8
( )b
af x dx Area of R1 – Area of R2 + Area of R3
Exemple1:
Integral Calculus
1/ Calculate algebraically the integral
2/ Use geometry to
compute the same integral
9
2
3
2
1
2
4
2
21
2
22
2
1
2
22
1
xxdx
Example2:
2
3
2
12
2
1
xdx
Some rules of integration
To simplify the determination of antiderivatives we can use the following rules. 1/ 2/ 3/ 4/ 5/ 6/
10
cxdx ckxkdx
cn
xdxx
nn
1
1
cxdxx
ln1
c
b
bdxb
xx ln
cedxe xx
11
Some rules of integration
7/ 8/ 9/ 10/
11/
1
1( 1)
nn ax b
ax b dx C na n
1 1
lnax b dx ax b Ca
1ax b ax be dx e Ca
1
ln
ax b ax bc dx c Ca c
f g dx fdx gdx
12
More examples
1/ 2/
3/ 4/ =
=
=
4 43 32 2 2
4 2
x xx dx x dx C C
cyydyyy 265
2
3)36(
cexdxex
xx
22
2
1log
1
dxx 2)16( dxxx )11236( 2
cxxx 23
2
12
3
36
cxxx 23 612
Examples 1/
2/ 4/
5/
6/
13
1
1
2 127 dxxx
2
0
2 33 dxx
3
0
dxe x
1
0
32 dxe x
5
1
11
2 dxx
x
Area Between Two Curves Let f and g be continuous functions, the area bounded above by f (x) and below by g(x) on [a, b] is:
14
b
a
dxxgxfR
a b
xfy
xgy
y
x
Area Between Two Curves
Find the area bounded by the curves where
and
R =
15
2
0
dxxgxfR
2xxf
2 xxg
Example:
R=14/3
1. By reversing the process of differentiation, we find
the original function from the derivative. We call
this operation integration or anti-differentiation.
2. The indefinite integral of a function is a function
defined as :
3. If f is a continuous function, the definite integral of