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Section 6.1: Areas between curves
Instructor: Ms. Hoa Nguyen
(nguyen@scs.fsu.edu)
The Area Problem
Special Case
:
Find the area of the region
S
that lies between two curves
y
=
f
(
x
) and
y
=
g
(
x
) from
a
to
b
(
f
,
g
are
continuous functions and
f
(
x
)
≥
g
(
x
)
for all
x
in
[
a, b
]). This means that
S
is bounded by the graphs
of continuous functions
f
,
g
, the vertical lines
x
=
a
and
x
=
b
.
The method
Divide
S
into
n
strips of equal width
4
x
=
b

a
n
. Then we have
n
subintervals: [
x
0
, x
1
]
,
[
x
1
, x
2
]
,
[
x
2
, x
3
]
,
· ·
·
,
[
x
n

1
, x
n
] where
x
0
=
a
and
x
n
=
b
.
Approximate the
i
th strip by a rectangle with base
4
x
and height
f
(
x
*
i
)

g
(
x
*
i
). The sample points
x
*
i
can
a right endpoint, left endpoint or any point in the
i
th subinterval [
x
i

1
, x
i
].
The
Riemann sums
are
R
n
= (
f

g
)(
x
1
)
4
x
+ (
f

g
)(
x
2
)
4
x
+
· · ·
+ (
f

g
)(
x
n
)
4
x
= Σ
n
i
=1
(
f

g
)(
x
i
)
4
x
if the right endpoints are used.
L
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 Fall '08
 Noohi
 Calculus

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