5 Steps to a 5 AP Calculus BC 2019

Definite Integrals 233

Example

Evaluate

∑n

i= 1

i(i+1)

n

.

Rewrite:

∑n

i= 1

i(i+1)

n

as

1

n

∑n

i= 1

(i^2 +i)=

1

n

( n

∑

i= 1

i^2 +

∑n

i= 1

i

)

=

1

n

(

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

6

+

n(n+1)

2

)

=

1

n

[

n(n+1)(2n+1)+ 3 n(n+1)

6

]

=

(n+1)(2n+1)+3(n+1)

6

=

(n+1)[(2n+1)+ 3 ]

6

=

(n+1)(2n+4)

6

=

(n+1)(n+2)

3

.

TIP • Remember: In exponential growth/decay problems, the formulas are dy
dx

=kyand

y=y 0 ekt.

Definition of a Riemann Sum

Let f be defined on [a,b] andxibe points on [a,b] such thatx 0 =a,xn=b, anda <

x 1 <x 2 <x 3 ···<xn− 1 <b. The pointsa, x 1 , x 2 , x 3 ,...xn+ 1 , andbform a partition of

f denoted asΔon [a,b]. LetΔxibe the length of theith interval [xi− 1 ,xi] andcibe any

point in theith interval. Then the Riemann sum off for the partition is

∑n

i= 1

f(ci)Δxi.

Example 1

Letf be a continuous function defined on [0, 12] as shown below.

x 0 2 4 6 8 10 12

f(x) 3 7 19 39 67 103 147

Find the Riemann sum for f(x) over [0, 12] with 3 subdivisions of equal length and the

midpoints of the intervals asci.

Length of an intervalΔxi=

12 − 0

3

=4. (See Figure 11.1-1.)