Area Between Curves

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Area between curves
Consider the region S
bounded by the graphs
of
y
=
f
(
x
),
y
=
g
(
x
)
and
the lines
x
=
a
and
x
=
b
.
The graphs of
f
and
g
are
continuous on [
a, b
].
( )
( )
f x
g x
Partition [
a
,
b
] into
n subintervals of equal
width
and
endpoints
Create corresponding
rectangles using, for
instance, the left end
points.
x
Area of S is roughly
the sum of the areas
of the rectangles:
1
0
(
)
(
)
Riemann Sum
n
i
i
i
A
f
x
g x
x
0
1
2
,
,
,
,
n
a
x
x
x
x
b

Area Between
Curves
NOTES:
The graph of
f
must lie above the graph of
g
on the interval [
a, b
].
As
n
increases, the approximation becomes better and better, thus we can define the
area of S as:
1
0
(
)
(
)
( )
( )
lim
n
b
i
i
a
n
i
A
f x
g x
x
f x
g x
dx

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