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6.1: Areas Between Curves
(Dated: September 26, 2011)
The area
A
of the region bounded by the curves
y
=
f
(
x
),
y
=
g
(
x
), and the lines
x
=
a
and
x
=
b
, where
f
and
g
are
continuous is given by
A
=
Z
b
a

f
(
x
)

g
(
x
)

dx
.
This can be obtained by considering the deﬁnition of the
integral on the righthand side in terms of Riemann sums.
Note that
A
=
Z
b
a
[
f
(
x
)

g
(
x
)]
dx
if
f
(
x
)
≥
g
(
x
) for all
x
in [
a
,
b
].
Likewise, the area
A
of the region bounded by the curves
x
=
f
(
y
),
x
=
g
(
y
), and the lines
y
=
c
and
y
=
d
, where
f
and
g
are continuous is given by
A
=
Z
d
c

f
(
y
)

g
(
y
)

d y
.
Example 1 (Selected exercises from exercises 6.1.56.1.28)
Sketch the region enclosed by the given curves.
Decide
whether to integrate with respect to x or y. Draw a typical
approximating rectangle and label its height and width.
Then ﬁnd the area of the region.
6: y
=
sin
x, y
=
e
x
, x
=
0
, x
=
π
2
Ans:
e
π
/2

2
8*: y
=
x
2

2
x,
y
=
x
+
4
Hint:
The curves
intersect when
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 Fall '09
 GELINE
 Calculus, Geometry

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