January 5, 2018

5.2 Riemann Sums and Definite Integrals

Sigma Notation

fete( tz ) ott .- - + fctn ) it

→

term

E aid,¥Ea€

.at#an0

-

-

y Ak → Ktn term

�5�K= index of summation

t.ae?k#inFeFrtakyEsnEheIYtima,µ

Lower lim ,

to the upper limit ofsummation ( n )

January 5, 2018

Sigma notation examples

µ§K= I +2+3+4+5=15

£' fifkk = - 1+2-3=-2;&t¥=ia+÷iEt¥?

622£2,2→ 1 +3+5+7+9 e-

�5�E¥a⇐II,k£1f.2 (1) + b= 1

*In Ztb =/

b= - 1

Ak =1

,3

, 5,7 ,9

[

=ai*aaT←

4

L EL1 +3+5+7+9

5

=EZK - 1

÷1

EZKHk¥fk+b=_)

2C of + b= 1 L

,b= 1

2f3)tb=1

{ 2kt 7

k= -3

January 5, 2018

Algebra rules for finite sums

England.)=§l,ak±§.

,b . fifteens

N n constant→ [ Can = C. Ea .

CER multiple

k=l K=lsn ←# of terms

→ (§C=Ctct . . . +c=n .c constant

K=l - Value

an terms

Fg3k-tiaEe3k.EIk2@ns3EIiE.k'

=3(1+2+3)

- (1+4+9)18 - ( 1 4 ) = @

n = 18

Sumoflst n -

integertoN= 1 - 1 •

2=1.2

z'

N=2 - 3 : .

6=23n=3-6µg. 12=3.4

N=y - 10.

:-..

.

20=4.5

5

n=5 - 15.:iI

.

30=5.6

# -.

'. . . a

2h - l

n(n÷

January 5, 2018

The sum of the first n...

Integers:

Squares:

Cubes:

+§k=ncz#

§.

,K2=Mn+ifRn+k§k3=( nay )2

1 7

n(ntl)§3 K - K 2 z

= 3 §?,

K - { K2

k = 1

=3 (my ) . (Easy= 3 ( 1 5 3)- §1 ) ( 3 5) )= 459 - 17 8 5

= - 1 3 2 6

SequenceI

, 3,5 ,7

,9

SeriesI +3+5+7+9

HW P.

292

1,5 , 10,11-16 ,18,21 , 26,28

January 5, 2018

Riemann Sums

January 5, 2018

January 5, 2018

January 5, 2018

Find the distance traveled by the projectile with the following velocity function, from 0 to 3 seconds. (We did this already)

January 5, 2018

January 5, 2018

January 5, 2018

Start by sketching

January 5, 2018

January 5, 2018

Find the area of the shaded region

January 5, 2018

Constant functions

January 5, 2018

January 5, 2018