5 Steps to a 5 AP Calculus BC 2019

Definite Integrals 235

Example 1

Use a midpoint Riemann sum with three subdivisions of equal length to find the approxi-
mate value of

∫ 6

0 x

(^2) dx.
Δx=

6 − 0

3

=2, f(x)=x^2

midpoints arex=1, 3, and 5.

∫ 6

0

x^2 dx≈ f(1)Δx+f(3)Δx+f(5)Δx=1(2)+9(2)+25(2)

≈ 70

Example 2

Using the limit of the Riemann sum, find

∫ 5
13 xdx.
Usingnsubintervals of equal lengths, the length of an interval

Δxi=

5 − 1

n

=

4

n

;xi= 1 +

(

4

n

)

i

∫ 5

1

3 xdx= lim

maxΔxi→ 0

∑n

i= 1

f(ci)Δxi.

Letci=xi; maxΔxi→ 0 ⇒n→∞.

∫ 5

1

3 xdx=nlim→∞

∑n

i= 1

f

(

1 +

4 i

n

)(

4

n

)

=nlim→∞

∑n

i= 1

3

(

1 +

4 i

n

)(

4

n

)

=nlim→∞

12

n

∑n

i= 1

(

1 +

4 i

n

)

=nlim→∞

12

n

(

n+

4

n

[

n

(

n+ 1

2

)])

=nlim→∞

12

n

(n+ 2 (n+ 1 ))=lim

n→∞

12

n

( 3 n+ 2 )=lim

n→∞

(

36 +

24

n

)

= 36

Thus,

∫ 5

1

3 xdx=36.

Properties of Definite Integrals

1. Iffis defined on [a,b], and the limit lim
   maxΔxi→ 0

∑n

i= 1

f(xi)Δxiexists, thenfis integrable

on [a,b].

2. Iff is continuous on [a,b], then fis integrable on [a,b].

Iff(x),g(x), andh(x) are integrable on [a,b], then