5 Steps to a 5 AP Calculus BC 2019

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

Figure 11.1-1

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 11.1-2.)

Figure 11.1-2

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.