4009 Fundamental Theorem of Calculus
(Part 2)
BC CALCULUS
The Indefinite Integral (Antiderivative) finds
a Family of Functions whose derivative is given.
A( x) 
cos(
t
)
dt

Given an Initial Condition we find the Particular Function

f( )3
2
The Definite Integral as a Particular Function:
Evaluate the definite integral.
x
A( x ) 
cos(
t
)
dt

0
Evaluate at
x  0,
  
,
,
6 4 3
Evaluate the Definite Integral for each of these points.
The Definite Integral is actually finding points on the Accumulation graph.
x
A( x ) 
 cos(t )dt
0
A( x)  sin( x)
Since A(x) is a function, what then is the rate of change of
that function?
A( x )  cos( x)
In words, integration and differentiation are inverse operations
2nd Fundamental Theorem of Calculus
x
Given:
A( x)   f (t )dt,
we want to find
A/ ( x )
a
2nd Fundamental Theorem of Calculus:
If f is continuous on an open interval, I, containing a point, a,
then for every x in I :

d 
  f (t )dt   f (u ) u
dx  a

u
Note:
a is a constant, u is a function of x;
and the order matters!
Demonstration: < function
x only >
x
A( x ) 
  sin(t )dt
find
d
( A( x))
dx
2
x

d
d 

( A( x)) 

sin(
t
)
dt
]



dx
dx 
 2

In Words:
Example:
Find and verify:
 x2  1

d 
2
   t  1 dt 
dx  0

x
Example:
Find without Integrating:
d 

dx  0
x
t
2

 1dt 

THE COMPOSITE FUNCTION
If g(x) is given instead of x:
d 
/
Q ( g ( x)) 

dx 
g ( x)

a

f (t ) dt 

d
[ F (t )]ag ( x )
dx
d
 F ( g ( x))  F (a) 
dx
=
F ( g ( x))* g '( x) or
f ( g ( x))* g '( x)
In words: Substitute in g(x) for t and then multiply by the
derivative of g(x)…exactly the chain rule
(derivative of the outside * derivative of the inside)
THE COMPOSITE FUNCTION
If
u  g ( x)
then
In Words:
, (a composite function)
u


d
/
f
(
t
)
dt

f
(
u
)
*
u


dx  a

Demonstration:
< The composite function >
cos( x3 )*(3x 2 )
Find:
 x3

d 

cos(
t
)
dt



dx 


4

In Words:
Example :
Find without Integrating:
x2
If
1
Q( x)   2 dt
t
2
,
solve for
/
Q ( x)
Example: Rewriting the Integral
dy
Find
without integrating: Show middle step
dx
5

x2
(2t  5) dt
Example: Rewriting the Integral - Two variable limits:
Find without Integrating:
d
dx

 

 cos( x )
sin( x )
break into two parts . . . . .
chose any number in domain of
.

t  1dt 


t 1
for a and rewrite into required form
Last Update:
• 1/25/11
• Worksheet