Definite Integral: Properties and Evaluation - Calculus Study Guide, Slides of Differential and Integral Calculus

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ANTIDERIVATIVES

(INTEGRAL)

THE DEFINITE INTEGRAL

• define and interpret definite integral,
• identify and distinguish the different

properties of

the definite integrals; and

• evaluate definite integrals

OBJECTIVES:

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 ,

( ) ( )

1. ,

 

 



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

b

a



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

 f x f x f x  dx f x dx f x dx f x dx

n

b

a

b

a

b

a

n

3. ( ) ( ).... ( ) ( ) ( ) ...... ( )  1 2  1 ^2

    

  

 

 

b

c

b

a

c

a

f x dx f x f x dx

where a c b then

If f x is continuous function in the closed erval a b

( ) ( ) ( )

, ,

4. ( ) int. ,



1

1

2

1. x dx

EXAMPLE:



1

0

3

2. (4x 2x 10)dx.





9

1

3. y(3 y)dy.

EXERCISES:





1

0

1. ( 3 2 x ) dx









2

3

2

2. y ( y 1 ) dy

  

a

a t dt

0

2



 

5

2

4. ( 5 x 2 )( 7 x 5 ) dx





3

1

4

2 2

5. dm

m

m





1 / 2

1 / 2

7

6. ( 3 x 1 ) dx





2

0

2 3

7. y 1 y dy

2

nd solution

f(4) 4; (4,4)

f(0) 0; (0,0)

f(-2) 2; (-2 ,2)

(x, y)

f(x) x







let 

1

2 3 4

(4,4)

-2,

0

( 4 )( 4 ) 8

2

1

( 2 )( 2 ) 2

2

1

2

1

 

 

A

A

1 2

4

2

 

xdx A A

 



3

3

2. 1 xdx

1

st solution







   

  

 

1 if 1 - x 0, x 1

1 if 1 - x 0, x 1

1

x

x

x

   

10

2

20

2

1

2

3

2

15

2

1

2 2

x

x -

1 1 1

3

1

2

1

3

2

3

1

1

3

3

3

     







  







     



 

x

x

xdx xdx x dx

 



2

1

3

3. x xdx

   







 ^  

  

 

if x 0

if 0

3 3

3 3

3

x x x

x x x x

x x

 

 1   1  0

1 0

0

2

3

  

 

 

x x x

x x

x x

 

 1   1  0

1 0

0

2

3

  

 

 

x x x

x x

x x

SS :  - 1,0   1 , SS :  -  ,-1   0 , 1 

     

4

11

2

1

3

1

0

3

2

1

0

1

3 3



             

x xdx x xdx x xdx x xdx

 

 

2

5

5. x 3 5

6. 2 3 7

5

0

7

1

  

   



 

dx

x dx

INTEGRATION OF ODD AND EVEN FUNCT

 

Foroddintegers,  - x

Forevenintegers, -x ;

Re :

n n

n n

x

x

call

 



Function is said to beevenif f(-x)  f(x)forallxdomain of f

Function is said to beoddif f(-x)  f(x)forallxdomain of f

The graph of an even function is symmetric about t

The graph of an odd function is symmetric about th

Theorem:

1. If f is odd on [-a,a] then

( )  0  

a

a

f x dx

a

-a

1

R

2

R

1 2

0

0

    

area of R area of R

f x dx f x dx f x dx

a

a

a

a

0 becausethefunction isodd

1

3

3

2

3





 

dt

t

t

EXAMPLE

 

 

 







 

 



6

4

2

2

6

0

0

• 2 . ( ). ( ) . ( ). f(x)dx

2 , 0

2 , 0

( )

1. In each part,evaluate theintegral given that

b f x dx d f x dx

a f x dx c

x if x

x if x

f x

EXERCISES