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Definite Integral - Engineering Mathematics, Lecture notes of Engineering Mathematics

FEU Institute of Technology (FEUIT)Engineering Mathematics

All the topics that the Definite Integral will cover. Unlock it and see it yourself; I will ensure it will help you.

Typology: Lecture notes

2021/2022

Available from 08/29/2022

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If F(x) is the integral of f(x)dx, that is, F’(x) = f(x)dx
and if a and b are constants, then the definite
integral is:
)a(F)b(F
xF dx)x(f
b
a
b
a
where a and b are called lower and upper limits of
integration, respectively.
The definite integral link the concept of area to
other important concepts such as length, volume,
density, probability, and other work.
THE DEFINITE INTEGRAL
pf3
pf4
pf5
pf8
pf9

Partial preview of the text

Download Definite Integral - Engineering Mathematics and more Lecture notes Engineering Mathematics in PDF only on Docsity!

If F(x) is the integral of f(x)dx , that is, F’(x) = f(x)dx

and if a and b are constants, then the definite

integral is:

F(b) F(a )

f ( x )dx F x

b a b a

where a and b are called lower and upper limits of

integration, respectively.

The definite integral link the concept of area to

other important concepts such as length, volume,

density, probability, and other work.

THE DEFINITE INTEGRAL

provided f x is defined in the closed erval  a b 

f x dx f x dx If a b then b a a b ( ) int ,

 

2. ( ) int ( ),

f x dx provided f a and f b exists

If a b and F x is the egral of f x then

ba

PROPERTIES OF DEFINITE INTEGRAL

( )  ( )  ( )  ( )  ( ) ( ) 0.

f x dx F x C F a C F a C F a F a

That is

b a b a

To obtain the definite integral of a function,

evaluate first its indefinite integral. Then applying

the limits of integration, that is, substitute the

upper limit of integration to all the variables

contained in the indefinite integral, minus the

function value of the indefinite integral using the

lower limit of integration.

    10

(4x 2x 10)dx x x 10x 4 2 4 2 1 0 1 0 3 4 2 

2 5 2 5 2 5 2 3 9 1 

  . 5

2y

3y y(3- y)dy (3y-y )dy 2 2 2 9 1 9 1

INTEGRATION OF ABSOLUTE VALUE FUNCTION

EXAMPLE^1.^ x dx

4  2

x

2 2 4 0 2 0 2 2 4 0 0 2 4 2

    

x x

dx xdx xdx

if x 0 if x 0 x 

x x -2 -1 0 1 2 3 4 x  0 x  0

 5 2 x- 3 dx. 5 3 5 2 2 3 2 2 3 2 By definition, x 3 if x 3 x 3 x 3 if x 3 Using property (v) of definite integral, x - 3 dx ( x 3)dx (x 3)dx -x x 3x 3x 2 2     ^ ^    ^ ^                     5 3 -9 25 9 9 ( 2 6) 15 9 2 2 2 25 2 2      (^)     (^)                        ^ ^   ^ ^ ^    29 2 