N=n => width = 1/n US = =1/n 3 [1+4+9+….+n 2 ] US =

..

N=n => width = 1/nN=n => width = 1/n

US = =1/nUS = =1/n33[1+4+9+….+n[1+4+9+….+n22]]

US = US =

2

1

1( )

n

i

i

n n

3 22

3 31

1 2 3

6

n

i

n n ni

n n

2

1

n

i

i

3 22 3

6

n n n

==

3 2

3

2 3lim

6n

n n nArea

n

3 2

3 3 3

3

3

2 3

lim6x

x x xx x x

xx

2

3 12

lim6x

x x

1

3

3 2

3

2 3lim

6n

n n n

n

0

DefiniteDefiniteIntegralIntegral

We will defineWe will define

to be the limit as nto be the limit as n

approaches oo ofapproaches oo of

where where xxii = (b-a)/n = (b-a)/n

and is any point in the ith interval. and is any point in the ith interval.

( )b

af x dx

*

1

( )n

i ii

f x x

*ix

*Area height width

where where xxii = (b-a)/n = (b-a)/n

and is any point in the iand is any point in the ithth interval, [x interval, [xi-i-

11,x,xii]. ].

*ix

*

1

( ) lim ( )nb

i ia ni

f x dx f x x

> 0 when f(x) > 0, > 0 when f(x) > 0, but it is but it is negative when f(x)<0.negative when f(x)<0.

We will defineWe will define

to be the limit as nto be the limit as n

approaches oo ofapproaches oo of

and is any point in the iand is any point in the ithth interval, [x interval, [xi-i-

11,x,xii]. ].

( )b

af x dx

* *

1 1

( )( ) ( )

n n

i i ii i

b af x x f x

n

*ix

( )b

af x dx

DefinitionDefinitionTheoremsTheorems

1. 1.

2. 2.

3.3.

4.4.

*

1

( ) lim ( )nb

i ia ni

f x dx f x x

( ) ( )a b

b af x dx f x dx

( ) 0a

af x dx

( ) ( )b b

a akf x dx k f x dx

( ) ( ) ( ) ( )b

b b

a aa

f x g x dx f x dx g x dx

DefinitionDefinitionTheoremsTheorems

5. 5.

6. 6.

7.7.

8.8.

( ) ( ) ( )b c c

a b af x dx f x dx f x dx

min ( )[ ] ( ) max ( )[ ]b

af x b a f x dx f x b a

( ) ( ) [ , ] ( ) ( )b b

a af x g x on a b f x dx g x dx 0 ( ) [ , ] 0 ( )

b

ag x on a b g x dx

*

1

( ) lim ( )nb

i ia ni

f x dx f x x

3

1

( )f x dx

1

3

( ) 2.3f x dx 4

3

( ) 2f x dx 4

1

( ) 6g x dx 4

1

( )f x dx 4

1

( ) ( )f x g x dx 4 4 4

1 1 1

2 ( ) 3 ( ) 2 ( ) 3 ( )f x g x dx f x dx g x dx

2.3 0.3

6.3

2(.3) 3(6) 18.6

If f(x) >= 0 on [a,b]If f(x) >= 0 on [a,b]

then is the area under f(x) then is the area under f(x) and and

over the x-axis between a and b. over the x-axis between a and b.

( )b

a

f x dx

If f(x) <= 0 on [a,b]If f(x) <= 0 on [a,b]

then is the negative of the then is the negative of the area area

over f(x) and under the x-axis over f(x) and under the x-axis between between

a and b. a and b.

( )b

a

f x dx

[[

0.500.50

0.10.1

1

0

( )f x dx

[[

2.02.0

0.10.1

1

0

4 ( )f x dx

]]

0.00.0

0.10.1

2

2

( )f x dx

[[

1.01.0

0.10.1

2

1

( )f x dx

3

2

( ) 0f x dx

[[

1.51.5

0.10.1

3

0

( )f x dx

00

2

0

sin( )x dx

]]

1.01.0

0.10.1

1

1

| |x dx

where where xxii = (b-a)/n = (b-a)/n

and is any point in the iand is any point in the ithth interval, [x interval, [xi-i-

11,x,xii].].

If the interval is [-4, 4] evaluate If the interval is [-4, 4] evaluate

88

*ix

*2

1

lim 16n

i ink

x x

4

2

4

16 x dx

*

1

( ) lim ( )nb

i ia ni

f x dx f x x

Pi = Pi = 3.143.14

6.286.28

0.10.1

22

2

4 x dx

Pi = Pi = 3.143.14

-6.28-6.28

0.10.1

22

2

4 x dx