Transcript MTH 251 Differential Calculus

MTH 252
Integral Calculus
Chapter 6 – Integration
Section 6.6 – The Fundamental
Theorem of
Calculus
Copyright © 2005 by Ron Wallace, all rights reserved.
Indefinite vs. Definite Integrals


f ( x )dx
b
a
f ( x)dx
is the set of functions of the
form F(x) + c where F’(x) = f(x)
n
is the number
lim
max xk 0
 lim
n 
*
f
(
x
 k )xk
k 1
n
 f ( x ) x
k 1
k
How are these two entities related?
Reminder: The Mean-Value Theorem
If f(x) is differentiable over (a,b), then there is a c(a,b)
where
f ( b)  f ( a )
f '( c ) 
ba
OR
f '(c)(b  a )  f (b)  f (a )
Reminder: The Mean-Value Theorem
If F(x) is differentiable over (xk-1,xk ), then there is a xk*(xk-1,xk )
where
F ( xk )  F ( xk 1 )
F '( x ) 
xk  xk 1
*
k
OR
F '( xk* )( xk  xk 1 )  F ( xk )  F ( xk 1 )
Reminder: Riemann Sum
n
*
*
*
*
f
(
x
)

x

f
(
x
)

x

f
(
x
)

x

...

f
(
x
 k k
1
1
2
2
n ) xn
k 1
where xk*  ( xk 1, xk ) and xk  xk  xk 1
If F’(x) = f(x) …
f ( x1* ) x1  F '( x1* )( x1  x0 )  F ( x1 )  F ( x0 )
f ( x2* ) x2  F '( x2* )( x2  x1 )  F ( x2 )  F ( x1 )
…
f ( xn* ) xn  F '( xn* )( xn  xn 1 )  F ( xn )  F ( xn 1 )
MVT:
Adding these
up gives …
F ( xn )  F ( x0 )
 F (b)  F ( a )
F '( x )( xk  xk 1 )  F ( xk )  F ( xk 1 )
*
k
Reminder: Definite Integral
If f(x) is continuous [a,b] and F’(x) = f(x) …

b
a
f ( x )dx 
n
lim
max xk 0
*
f
(
x
 k )xk
k 1
F (b)  F (a )

max x 0
 lim
k
 F (b)  F ( a )
Fundamental Theorem of Calculus
(Part I)
If f(x) is continuous [a,b] and F’(x) = f(x) …

b
a
f ( x)dx  F (b)  F (a )
 F ( x )a
b
b
  f ( x )dx 
a
Note that this works with ANY antiderivative. Why?
Integrating a Piecewise Function
Evaluate

5
0

5
0
 x2
f ( x )dx where f ( x )  
6  x
2
5
0
2
f ( x)dx   f ( x)dx   f ( x)dx
  x dx    6  x  dx
2
2
0
5
2
2
5
x  
x 
    6 x  
3 0 
2 2
3
2
8 
25 
61
   30    12  2  
3 
2 
6
x2
x2
Integrating an Absolute Value Function
Evaluate

3
1
4  x dx  
2
2
1
2

3
1
4  x 2 dx
2
4

x
dx

x

    4 dx
3
2
2
3

x

x 
  4 x      4 x 
3   1  3

2
3
3
8 
1    27

 8

  8     4       12     8    12
3 
3    3
  3 

Mean Value Theorem for Integrals
Let f(x) be continuous over [a,b].
Extreme-Value Theorem   m & M where
m ≤ f(x) ≤ M
Therefore …

b
a
b
b
a
a
mdx   f ( x)dx   Mdx
Or …
b
m(b  a)   f ( x)dx  M (b  a )
a
Mean Value Theorem for Integrals
Therefore …
1 b
m
f ( x )dx  M

ba a
Intermediate-Value Theorem  f(x) takes on
all values between m & M. Remember: m  f(x)  M
Therefore, there exists x*[a,b] where …
b
b
1
m(b  fa() x ) f ( x)dx f M
()bdx a )
(
x
a
ba a
*
Mean Value Theorem for Integrals
If f(x) is continuous over [a,b], then there exists
x*[a,b] where …
1 b
f (x ) 
f ( x )dx

ba a
*
f(x*)
OR

b
a
f ( x)dx  f ( x )(b  a )
The number
*
a x*
b
f(x*) is called the average value of the function.
The Area Function
x
A(x)
A( x)   f (t )dt
a
d
d  x

A
(
x
)

f
(
t
)
dt
?
 


dx
dx  a
The Derivative of the Area Function
x h
d  x

a

f
(
t
)
dt

lim
 h0
dx  a
x
f (t )dt   f (t )dt
a
h
1 x h
 lim  f (t )dt
h 0 h x
1
 lim  f (t * )h  where t *  ( x, x  h )
h 0 h
MVT for Integrals
 f ( x)
Fundamental Theorem of Calculus
(Part II)
If f(t) is continuous over an interval
containing a, then
d  x
  f ( x)
f
(
t
)
dt

dx  a
FTC Part II: Examples
d  x1 
1
dx   2
2


dx  1 t
x

d 

2
sin tdx   2 x sin  x 


dx  1

x2
NOTE: Chain Rule!
Check these out by
integrating and then
differentiating!