Fundamental Theorem
AP Calculus
Where we have come.
Calculus I:
Rate of Change Function
f f
y
f’
f
x
T
P
D
D
C
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
Accumulation: Exact Accumulation
Vy
x 
8
n
xi 
8i
n
 8i 
f ( xi )  f  
n
 8i   8 
lim  f    
n 
 n  n 
i 1
n
f (xi )
x
Tx
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:
a)
b)
T
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
lim  f (ci ) xi
x 0
i 1
exists , then
b


a
f ( x) dx
B). The Fundamental Theorem
The Definition of the Definite Integral shows the set-up.
Your work must include a Riemann’s sum! (for a representative rectangle)
n
lim  f (ci ) xi
x 0
n
i 1
lim  f (ci ) xi 
x  0
i 1
b
 f ( x)dx
a
The Fundamental Theorem of Calculus (Part A)
If
F ( x)  f ( x)
or
F
is an antiderivative of
b
then

a
f ( x )dx


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!
b

a
f ( x )dx 
F  x   a
b
= F b  F  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

2).
(
x

1)
dx


3).   sin( x) dx
1)
3
4
xdx
2
1
2

2
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
interval [-2,3].
REQUIRED:
Your work must include a Riemann’s
sum! (for a representative rectangle)
on the
Example: Work
Find the NET Accumulation represented by the region between
the graph and the x - axis
f ( x)  x  2 x  5
2
interval [-2,3].
REQUIRED:
Your work must include an antiderivative!
on the
Method: (Grading)
A).
1.
2.
B).
3.
4.
C).
5.
D).
6.
7.
Example:
Find the NET Accumulation represented by the region between
the graph and the x - axis
interval
0,3 .
f ( x)  27  x
3
on the
Example:
Find the NET Accumulation represented by the region between
the graph and the x - axis
 

interval   , 
 4 3
.
f ( x)  sec( x) tan( x) on the
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