4032 Fundamental Theorem
AP Calculus
Where we have come.
Calculus I:
Rate of Change Function
f f
y
f’
f
v(t)
x
T
P
D
+
-
P : f(0)=0
D
∪
∩
C
2.5
∪
6
8
T
Where we have come.
Calculus II:
Accumulation Function
Accumulation: Riemann’s Right
Vy
Tx
Accumulation (2)
Using the Accumulation Model, the Definite Integral represents
NET ACCUMULATION -- combining both gains and losses
D
y
V
5
8 8 6
3
T
x
-3
-4
-3
T
REM: Rate * Time = Distance
Where we have come.
Calculus I:
Rate of Change Function
f f
Calculus II:
Accumulation Function
Using DISTANCE model
f’ = velocity
f = Position
Σ v(t) Δt = Distance traveled
Distance Model: How Far have I Gone?
Vy
x
Distance Traveled:
T
a)
If I go 5 mph for one hour and 25mph for 3
hours what is the total distance traveled?
b)
Ending position-beginning position
B). The Fundamental Theorem
DEFN: THE DEFINITE INTEGRAL
If
f
is defined on the closed interval [a,b] and
n
lim  f (ci ) xi
x 0
n
i 1
Height
base
Rate
time
lim  f (ci ) xi
x 0
i 1
exists , then
b


f ( x) dx
a
The Definition of the Definite Integral shows the set-up.
Your work must include a Riemann’s sum! (for a representative rectangle)
The Fundamental Theorem of Calculus (Part A)
If
F ( x)  f ( x)
or
F
is an antiderivative of
b
then

f ( x )dx
a


f,
F  x   a
b
F b  F  a 
The Fundamental Theorem of Calculus shows how to solve the problem!
Your work must include an anti-derivative!
𝑣 𝑡 𝑑𝑡
 d (t )
b
a
𝑑(𝑏) − 𝑑(𝑎)
REM: The Definite Integral is a NUMBER -- the Net Accumulation
of Area or Distance -- It may be positive, negative, or zero.
Practice:
Evaluate each Definite Integral using the FTC.
1
𝑥2 1
1 −32
1)
=
= −
= −4
2 −3
2
2

2).
(
x

3
4
xdx
1

3).


Top-bottom
2
2
2
 1)dx
sin( x)dx
64
−1
𝑥3
4
=
−4 −
+1
=
−𝑥
3
3
3
−1
52 2 50
− =
3 3
3
𝜋
= −𝑐𝑜𝑠𝑥 2𝜋
−
2
= −0 + 0 = 0
The FTC give the METHOD TO SOLVE Definite Integrals.
Example: SET UP
Find the NET Accumulation represented by the region between
the graph and the x - axis
f ( x)  x  2 x  5
2
on the
interval [-2,3].
𝑏 = ∆𝑥 −2,3
ℎ = 𝑥 2 − 2𝑥 + 5-0
𝑡𝑜𝑝 − 𝑏𝑜𝑡𝑡𝑜𝑚
𝐴 = 𝑥 2 − 2𝑥 + 5 ∆𝑥
∆𝑥
𝑥 2 − 2𝑥 + 5 ∆𝑥
lim
𝑛→∞
REQUIRED:
Your work must include a Riemann’s
sum! (for a representative rectangle)
3
𝑥 2 − 2𝑥 + 5 ∆𝑥
−2
Example: Work
Find the NET Accumulation represented by the region between
the graph and the x - axis
interval [-2,3].
f ( x)  x  2 x  5
2
on the
𝑥3
3
2
− 𝑥 + 5𝑥
3
−2
27
−8
− 9 + 15 −
− 4 − 10
3
3
45 50 95
+
=
3
3
3
REQUIRED:
Your work must include an antiderivative!
∆𝑥
Method: (Grading)
A).
B)
1.
Graph and rectangle
2.
Base ∆𝑥 𝑜𝑟 ∆𝑦 and boundaries
3.
Height (top – bottom) or (right – left) or (big – little)
4.
Riemann’s Sum
5.
C).
Definite Integral [must have dx or dy]
6. antiderivative
7.
answer
Example:
Find the NET Accumulation represented by the region between
the graph and the x - axis
interval
f ( x)  27  x
0,3 .
3
on the
√
3
27 − 𝑥 3 𝑑𝑥
√
0
𝑥4 3
27𝑥 −
4 0
√
81
81 −
− 0−0
4
243
4
𝑏 = ∆𝑥 0,3 √
ℎ = 27 − 𝑥 3 − 0 √
𝐴 = 27 − 𝑥 3 ∆𝑥
3
√
27 − 𝑥 3 ∆𝑥 √
lim
𝑛→∞
0
Example:
Find the NET Accumulation represented by the region between
the graph and the x - axis
  
interval   , 
 4 3
√
𝜋
3
−𝜋
4
.
√
sec 𝑥 tan 𝑥 𝑑𝑥
√ sec 𝑥
√
f ( x)  sec( x) tan( x) on the
2−
2
2
𝜋
3
−𝜋
4
=
−𝜋 𝜋
𝑏 = ∆𝑥
,
√
4 3
√ ℎ = sec 𝑥 tan 𝑥 − 0
2 2−2
2
4−2 2
=
2
𝜋
3
lim
√ 𝑛→∞
𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 ∆𝑥
𝜋
−4
Last Update:
• 1/20/10
Antiderivatives
Layman’s Description:
2
x
 dx
 cos( x)dx
1
 x 2 dx
Assignment: Worksheet
 sec ( x)dx
2
1
 x dx
Accumulating Distance (2)
Using the Accumulation Model, the Definite Integral represents
NET ACCUMULATION -- combining both gains and losses
D
V
4
T
T
REM: Rate * Time = Distance
Rectangular Approximations
y = (x+5)(x^2-x+7)*.1
V = f (t)
Velocity
Distance Traveled:
a)
b)
Time