Fin118 Unit 4

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

f from a to b is defined as:

cxFdxxf

aFbFdxxfb

a

Time to Review !

16

we will see in the next unit

17

Matrix / Matrices

Different types of matrices

Usual operations on matrices