THE FUNDAMENTAL THEOREM OF CALCULUS
The Fundamental Theorem of Calculus is appropriately named because it establishes
connection between the two branches of calculus: differential calculus and integral
calculus.
DEFINITION
A function
F (x )is called an
antiderivative of f (x ) if
F ' ( x)  f ( x)
Example
Let:
f ( x)  x 2
Find the an antiderivative F (x )
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2

f ( x) cont on [a, b]
b
a
f ( x)dx  F (b)  F (a)
where F is any antideriva tive of f
Example:
Evaluate the integral

3
1
Example:
Find the area under the curve
x
e dx
y  x2
from x-0 to x=1
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2
f ( x) cont on [a, b]

b
a
f ( x)dx  F (b)  F (a)
where F is any antideriva tive of f
Example:
Evaluate the integral

6
3
Example:
1
dx
x

Find the area under the curve y  cos x from x-0 to x 
2
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 2

f ( x) cont on [a, b]
b
a
f ( x)dx  F (b)  F (a)
where F is any antideriva tive of f
Example:
Evaluate the integral
1
1 x 2 dx
3
THE FUNDAMENTAL THEOREM OF CALCULUS
Define:
x
F ( x)   f (t )dt
a
x
Example: F ( x)   f (t )dt
0
Find : F (0), F (1), F (2)
Example:
x
F ( x)   tdt
0
Find : F (0), F (1), F (2)
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1
f ( x) cont on [a, b]
1) F ( x) cont on [a, b]
x
F ( x)   f (t )dt
2) F ( x) diff on (a, b)
a
3) F ' ( x)  f ( x)
a xb
Example:
Find the derivative of the function
F ( x)  
x
0
Note:
1  t 2 dt
Using Leibniz notation
3) F ' ( x)  f ( x)
d x
f (t ) dt  f ( x)

a
dx
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1
f ( x) cont on [a, b]
x
F ( x)   f (t )dt
a
a xb
Note:
1) F ( x) cont on [a, b]
2) F ( x) diff on (a, b)
3) F ' ( x)  f ( x)
Using Leibniz notation
3) F ' ( x)  f ( x)
Example:
Find
d x
f (t ) dt  f ( x)

a
dx
d x
sec tdt

1
dx
THE FUNDAMENTAL THEOREM OF CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS, PART 1
f ( x) cont on [a, b]
1) F ( x) cont on [a, b]
x
F ( x)   f (t )dt
2) F ( x) diff on (a, b)
a
3) F ' ( x)  f ( x)
a xb
Note:
Using Leibniz notation
3) F ' ( x)  f ( x)
Example:
Note:
Find
d x
f (t ) dt  f ( x)

a
dx
d x4
sec tdt

dx 1
d h( x)
f (t )dt  f (h( x))  h' ( x)

dx a
THE FUNDAMENTAL THEOREM OF CALCULUS
Note:
Note:
d h( x)
f (t )dt  f (h( x))  h' ( x)

a
dx
d a
f (t ) dt 

h
(
x
)
dx
Note:
d h2 ( x )
f (t )dt  f (h2 ( x))  h'2 ( x)  f (h1 ( x))  h'1 ( x)

dx h1 ( x )
THE FUNDAMENTAL THEOREM OF CALCULUS
Note
d x
f (t ) dt  f ( x)

dx a
which says that if f is integrated and then the result is differentiated, we
arrive back at the original function
Note

b
a
F ' ( x)dx  F (b)  F (a)
This version says that if we take a function , first differentiate it, and then
integrate the result, we arrive back at the original function
Total Area
Example
Evaluate:

2

2
2
2
( x 2  4)dx
(4  x 2 )dx
Total Area
Total Area
Example
Find the area of the region between
the x-axis and the graph of
f ( x)  x 3  x 2  2 x, [1,2]
Example
Find the area of the region between
the x-axis and the graph of
2
3
2
(
x

x
 2 x) dx

1
What is the difference
between these two examples
TERM-102
TERM-102
TERM-092
TERM-082
TERM-082
TERM-082
TERM-091
TERM-093
TERM-091
Term-092
Term-082