J. Kim MS125 Section 5.3 (Nov. 6, 2006)
5.3 & 5.4 The Definite Integral and The Fundamental Theorem of Calculus
Properties of Definite Integrals I
1. If
0)( xf
for
x
in
],[ ba
,
dxxf
b
a
)(
Area under the graph of
)(xf
between
a
and
b
2. If
0)( xf
for
x
in
],[ ba
,
dxxf
b
a
)(
--(Area under the graph of
)(xf
between
a
and
b
)
Properties of Definite Integrals II
3.
dxxf
a
a
)(
4. If
bca
,
b
c
c
a
b
a
dxxfdxxfdxxf )()()(
5.
b
a
a
b
dxxfdxxf )()(
6. If
k
is a real number,
b
a
b
a
dxxfkdxxkf )()(
7.
b
a
b
a
b
a
dxxgdxxfdxxgxf )()()()(
8. If
)()( xgxf
on
],[ ba
, then
b
a
b
a
dxxgdxxf )()(
Example 1) Example 1 on page 265
Example 2) Suppose
16)(
7
0
dxxf
and
6)(
7
4
dxxf
.
Find a)
4
0
)( dxxf
and b)
4
7
)6)(2( dxxf
.
Example 3) Evaluate the definite integral
2
0
)31( dxx
.
Example 4) (a) Is
2
0
sin
dxx
positive, negative, or zero? Explain.
(b) Explain why
2cos0
2
0
dxx
Properties of Definite Integrals III
1. If
)(xf
is even, then
dxxfdxxf
aa
a
0
)(2)(
.
2. If
)(xf
is odd, then
0)(
dxxf
a
a
.
