Definite Integrals Notes and Examples - Riemann Sums

3
#*
2
##)&%045




>>>
?@
;
<
# 6 78 9


::
0




0




=



6
6 $% C DE"(" 6 )("
$& 6 $A/ 6 $ 6 AB 0 0 0 6 0 0 0 0 0 0




% 6 ("F&)%'*"+




0
G $


I&
H
) J




!"#$%$&"$%&"'()*+#*,##)-./0)1
K 6 DE" 6 +1)**"( 6 48

DE" 6 1>(" 6
("F&)%'*"+ 6 >% 6 $%&"(L)*
M E$'E"( 6 % 6 L)*N" 9

DE" 6 1>(" 6 )FFN()&" 6 &E" 6 )(")
$%&"(L)* +$/"
O
)GG(>8$1)&$>%
J
0




48 >#
6 ("F&)%'*"+
A
F- 6 %> 0 6 6




- -
-
6 0 6 - 6 - 6 - -




P
0 -
- - - - -



48
- -



8$A 6 )& 6 $0




A
6 )&$ M J$& 9 Q 7)*FN*N+
I

H

4(")>#$&ER"F&)%'0ISQ 6
=


4$ #*)& 6 48 9 S8)F& 6
D>&)* 4(") T%,"(
48
U()GE
-

6 $0
I




4GG(>8$1)*"->&)*E-(")N%,"()-()GE01"V4A%EW>>
- 0




-
J(")&E
I #

E"$'E& I




=
M ?I 9 #M )& 6
$MJ 9 9
9
I
4 6 A
0
48 0
W- M )&$ 0
48 X)

>E
3

$%#$%$&" 6 )1>N%& 6


I
9 #M $MJ 9 9
("F&)%'*"+
M J
H
- - - 0 0 6 -




)& 6 =


4*+>
A

=
=
6 Y($&"( 6 )+ H




Z - 0




H
6
4
')J#F)9 &E"1"
H+
\]S%S
S[Z
S()+"(+
A
=

F)J+
,
>%
6

=



M J$& 9 ##)&$#J%) 9 9
I J

S) S$
I ^&")* A




)
I

, R$"1)%% ^N%
-




]@
3
&"(1+
5>(" U"%"()* Z
$%&"'()&"
L)($)J*"
0

6
#
$%

0



&E" $&E




`aabc8AS#
6
_$ 6C 6 *$"+ 6 >% 6

\ I $%&"(L)*
$# +NJ6

0
S$ $

&E$+ F8$ 0 0
0
0F$ =


&E" *$1$&
d
I# 6 6 "8$+&+
=




# 6 $+ 6
*>Y"( ;6
NGG"(
Y" 6
+)X $%&"'()J*"
*$1$&+ 6 ># $%&"'()&$>% >% 6 &E" 6 $%&"(L)* e )
= JX




3
3 H
3
fE)& 3 3
6 Y" 6
N+N)**X N+"
Z

"& ' 3
3
R! 0
>
H


S 3
+> X(
'()L"
$&
O
# $%&"'()J*" e J g &E"%


HD
I#
H



$+ Z
6 6 >% ) = 6 ("F&)%'*"
C
,"#$%"
&E$+)()+"%&E">P-"0ID
I# a)$I#F
Y" 6
7. ,>F
-
-


0




O 6 A 6 ) 6 &)8
7$&$ 6 4+F O 6 8 =

YE"(" _$ 6
6
A

h/ 6 A 6 )& 6 i4h



)%, 48 6 A J j% 6 A 6 )& 6 %6 48


A 6 )& 6 % 0

J

A J