Exact Accumulation and

AP Calculus

A). Sigma Notation

REM:

6

3 3 3 3 3 3 3

1

1 2 3 4 5 6i

i

6

3

1i

i

7

3

2 1i

i

Ex.

Summation Formulas

1

n

i

c cn

2

1

12 2

n

i

n n ni n or

3 22

1

1 2 1 2 3

6 6

n

i

n n n n n ni or

22 4 3 23

1

1 2

4 4

n

i

n n n n ni or

REM: Sum of the first 100 numbers. 1+2+3+…….+98+99+100

Finite Series

EX 1. Evaluate using the formulas:

15

1

2 3i

i

EX 2. Evaluate using the formulas:

15

2

1

2i

i

Infinite Series

EX 3. Evaluate: 2

31

13

n

ni

Lim in

REM: Limits at Infinity

NOTICE: Inside:

i is the variable.

Outside: n is the variable

23

1

43 2

n

ni

Lim i xn

EX 4. Evaluate:

AREA

DEFN: The Area (accumulation) defined by a region in the plane is:

1

n

i in

i

A Lim f c x

1i i ix c x

We will use the RIGHT ENDPOINT and REGULAR PARTITIONS (easier!) b a

xn

if x

THEOREM: The Limit ( as ) of the Under Approximation and the Over Approximation (and therefore any Riemann’s Sum) are equal.

n

a b

a

x

( )if x

ith

ix

ith

Rect.

( )if x1st

Rect.2nd

Rect.

ix

Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded

by and the x - axis on the interval [ 0, 2].2y x

Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded

by and the x - axis on the interval [ 0, 3].24y x x

x

y

Area: Exact Accumulation Find the Area (accumulation) determined by the region bounded

by and the x - axis on the interval [ 1, 2].2( ) 4f x x

Area: Exact AccumulationFind the Area (accumulation) determined by the region bounded

by and the x - axis on the interval [ 2, 4].24y x x

x

y

Last Update:

• 1/15/10

• Assignment: Worksheet