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

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

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

2

c 1 c 2

x 1

x 0 = 0

y

x 2 x 3

x

c 3

4681012

Figure 12.1-1

Riemann sum=

∑^3

i= 1

f(ci)Δxi=f(c 1 )Δx 1 + f(c 2 )Δx 2 +f(c 3 )Δx 3

=7(4)+39(4)+103(4)= 596

The Riemann sum is 596.

Example 2

Find the Riemann sum for f(x)=x^3 +1 over the interval [0, 4] using 4 subdivisions of

equal length and the midpoints of the intervals asci. (See Figure 12.1-2.)

0.5 1 1.5 2 2.5 3 3.5 4

c 1 c 2

x 1

x 0 = 0

y

x 2 x 3 x 4

x

c 3 c 4

Figure 12.1-2

Length of an intervalΔxi=

b−a

n

=

4 − 0

4

=1;ci= 0. 5 +(i−1)=i− 0 .5.

Riemann sum=

∑^4

i= 1

f(ci)Δxi=

∑^4

i= 1

[

(i− 0 .5)^3 + 1

]

1

=

∑^4

i= 1

(i− 0 .5)^3 +1.

Enter

∑(

(1− 0 .5)^3 +1,i,1,4

)

= 66.

The Riemann sum is 66.

Definition of a Definite Integral

Letf be defined on [a,b] with the Riemann sum forfover [a,b] written as

∑n

i= 1

f(ci)Δxi.

If max Δxi is the length of the largest subinterval in the partition and the

lim

maxΔxi→ 0

∑n

i= 1

f(ci)Δxiexists, then the limit is denoted by:

lim

maxΔxi→ 0

∑n

i= 1

f(ci)Δxi=

∫b

a

f(x)dx.

∫b

a

f(x)dxis the definite integral of ffromatob.