6.7.ppt evaluating definite integral with AP Classroom activities

Evaluating Definite Integrals
Theorem: If f is continuous on the interval [a, b], then
where F is any antiderivative of f, that is, F’=f.
Example:
since F(x)=1/3 * x3
is an antiderivative of f(x)=x2
 

b
a
a
F
b
F
dx
x
f )
(
)
(
)
(
3
1
0
3
1
1
3
1
)
0
(
)
1
( 3
1
0
3
2







 F
F
dx
x

The Fundamental Theorem of
Calculus
The theorem establishes a connection between the two
branches of calculus: differential calculus and integral
calculus:
Theorem: Suppose f is continuous on [a, b].
Example of Part 1: The derivative of
is









b
a
x
a
f
F
f
F
a
F
b
F
dx
x
f
x
f
x
g
dt
t
f
x
g
is
that
,
of
tive
antideriva
any
is
where
),
(
)
(
)
(
.
2
)
(
)
(
then
)
(
)
(
If
.
1


x
dt
t
x
g
0
2
)
(
2
)
( x
x
g 


Average value of a function
The average value of function f on the interval [a, b] is
defined as
Note: For a positive function, we can think of this
definition as saying area/width = average height
Example: Find the average value of f(x)=x3
on [0,2].



b
a
ave dx
x
f
a
b
f )
(
1
2
4
2
2
1
4
2
1
0
2
1 4
2
0
4
2
0
3











 
x
dx
x
fave

The Mean Value Theorem
for Integrals
Theorem: If f is continuous on [a, b], then there exists a
number c in [a, b] such that
Example: Find c such that fave=f(c) for f(x)=x3
on [0,2].
From previous slide, f(c)=fave=2.
Thus, c3
=2, so
)
)(
(
)
(
is,
that
)
(
1
)
(
a
b
c
f
dx
x
f
dx
x
f
a
b
f
c
f
b
a
b
a
ave







26
.
1
2
3


c

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6.7.ppt evaluating definite integral with AP Classroom activities