5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:57

248 STEP 4. Review the Knowledge You Need to Score High

12.1 Riemann Sums and Definite Integrals

Main Concepts:Sigma Notation, Definition of a Riemann Sum, Definition of

a Definite Integral, and Properties of Definite Integrals

Sigma Notation or Summation Notation

∑n

i= 1

a 1 +a 2 +a 3 +···+an

whereiis the index of summation,lis the lower limit, andnis the upper limit of summation.

(Note: The lower limit may be any non-negative integer≤n.)

Examples

∑^7

i= 5

i^2 = 52 + 62 + 72

∑^3

k= 0

2 k=2(0)+2(1)+2(2)+2(3)

∑^3

i=− 1

(2i+1)=− 1 + 1 + 3 + 5 + 7

∑^4

k= 1

(−1)k(k)=− 1 + 2 − 3 + 4

Summation Formulas

Ifnis a positive integer, then:

1.

∑n

i= 1

a=an

2.

∑n

i= 1

i=

n(n+1)

2

3.

∑n

i= 1

i^2 =

n(n+1)(2n+1)

6

4.

∑n

i= 1

i^3 =

n^2 (n+1)^2

4

5.

∑n

i= 1

i^4 =

n(n+1)(6n^3 + 9 n^2 +n−1)

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