4028 The Fundamental Theorem (A)
Where we have come.
y
Calculus I:
Rate of Change Function
f f
f
x
P
D
D
C
Calculus II: Accumulation Function Reimann’s Right
Where we have come
y
f
x
Using the Accumulation Model, the Definite Integral represents NET ACCUMULATION -- combining both
gains and losses
Accumulation: Exact Accumulation
x 
8
n
xi 
 8i 
f ( xi )  f  
n
8i
n
n
 8i   8 
lim  f    
n 
 n  n 
i 1
Using DISTANCE model
f ’ = velocity
f = Position
Σ v(t) Δt = Distance traveled
Using DISTANCE model
Distance Traveled: Net (Displacement)
a)
or
b)
B). The Fundamental Theorem
DEFN: THE DEFINITE INTEGRAL
If f is defined on the closed interval [a,b] and
n
lim  f ( ci ) xi exists ,
x 0
i 1
n
then
lim  f (ci )xi
x 0
i 1
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 f,
b

then
f ( x )dx

F  x   a

F b  F a 
a
b
The Fundamental Theorem of Calculus shows how to solve the problem!
Your work must include an anti-derivative!
REM: The Definite Integral is a NUMBER -- the Net Accumulation of Area or Distance –
It may be positive, negative, or zero.
Evaluate each Definite Integral using the FTC.

1
xdx
3

4
1



1)
4)
( x 2  1)dx
2
sin( x )dx
2

3
1

8
1
x5dx
3
x dx
2)
5)
 1  2 x  4 x  dx
3
3
1
2

cos   d
3)
6)
 u
0
2



3
4
5
 u3  u2  du
sec( ) tan( )d
Example: Show the SET-UP
Find the NET Accumulation represented by the region between
the graph and the x - axis f ( x)  x 2  2 x  5 on the
interval [-2,3].
REQUIRED:
Your work must include a Riemann’s sum!
(for a representative rectangle)
Grading:
A).
C).
1. ___________________________ B).
3. ________________________________
2. ___________________________
4. ________________________________
5. ___________________________ D).
6. ________________________________
7. ________________________________
Example:
Find the NET Accumulation represented by the region between
the graph of f ( x)  27  x 3 and the x - axis on the
interval [ 0 , 3 ]
.
Assignment 4028
B. The Fundamental Theorem of Calculus
BY HAND – Show all steps to solve using the FTC.
Evaluate the definite Integral.
1
(1)

 (5 x  14)dx
(2)
4

(3)
(5)
(7)

4
sin(t )dt
(4)
3

4

2
2
0
1
dx
x
(6)
e x dx
(8)


3
cos( x) dx
2
1
dy
y4


1

e
1
2
dx
x

2

csc 2 (t )dt
6
Calculator – Show all steps to the set-up using the DEFN of DEFINITE INTEGRAL
Find the NET accumulation of the function with respect to the x-axis.
f ( x)  3x 2  x  2
x   0,3
(10)
f ( x)  x  2 x
x  0,4
(11)
f ( x)  2 x  cos( x)
  
x   , 
 2 2
(12)
f ( x) 
(9)
x2
x
x 1,4