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27. Area Between Curves
P. K. Lamm
11/27/11 (11:53)
p. 1 / 13
Lecture Notes:
27. Area Between Curves
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of deﬁnitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. The Area between Two Curves
Suppose we have two continuous curves
y
=
f
(
x
) and
y
=
g
(
x
) where
f
(
x
)
> g
(
x
) for all
x
∈
[
a,b
].
The region bounded by these two curves and by the lines
x
=
a
and
x
=
b
is given in the graph
below:
To evaluate this area, we will compute a Riemann sum using right endpoints. Letting
Δ
x
=
b

a
n
,
we will partition the interval [
a,b
] into
n
equal intervals (each of length Δ
x
) by deﬁning endpoints
x
0
=
a
,
x
1
=
a
+ Δ
x
,
x
2
=
a
+ 2Δ
x
, etc.
. That is,
x
k
=
a
+
k
Δ
x
,
k
= 0
,
1
,...,n
.
The
n
right endpoints are given by
c
k
=
x
k
=
a
+
k
Δ
x
k
= 1
,
2
,...,n.
1
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View Full Document 27. Area Between Curves
P. K. Lamm
11/27/11 (11:53)
p. 2 / 13
The
k
th
rectangle has its top sitting at a point on the upper curve
y
=
f
(
x
) and its base sitting
at a point on the lower curve
y
=
g
(
x
). That is, the height of the
k
th
rectangle is given by
height of
k
th
rectangle =
f
(
c
k
)

g
(
c
k
)
.
The area of the
k
th
rectangle is thus (
f
(
c
k
)

g
(
c
k
))Δ
x
, and the combined area of all
n
rectangles
may be written as
area of
n
rectangles =
n
X
k
=1
(
f
(
c
k
)

g
(
c
k
)) Δ
x
The actual area of the region is then given by
actual area = lim
n
→∞
n
X
k
=1
(
f
(
c
k
)

g
(
c
k
)) Δ
x
or
the area between
y
=
f
(
x
),
y
=
g
(
x
),
x
=
a
, and
x
=
b
(
f
(
x
)
> g
(
x
) on [
a,b
])
=
Z
b
a
(
f
(
x
)

g
(
x
))
dx.
Note:
This idea generalizes our notion of the area “under a curve”; i.e., the area between the
curve
y
=
f
(
x
) (
f
(
x
)
>
0,
x
∈
[
a,b
]), the
x
axis, and the lines
x
=
a
and
x
=
b
. We may now
view this region as that found
between
the curves
y
=
f
(
x
) and
y
= 0,
a
≤
x
≤
b
.
Thus in this case,
the area between
y
=
f
(
x
), the
x
axis,
x
=
a
, and
x
=
b
(
f
(
x
)
>
0 on [
a,b
])
=
Z
b
a
(
f
(
x
)

0)
dx
=
Z
b
a
f
(
x
)
dx.
Example 1.1:
Find the area bounded by the lines
y
=
x
,
x
= 2 and the curve
y
= 1
/x
2
.
2
27. Area Between Curves
P. K. Lamm
11/27/11 (11:53)
p. 3 / 13
Our analysis makes use of the following facts:
1. The “upper curve” for the region is:
y
=
f
(
x
) =
x
.
2. The “lower curve” for the region is:
y
=
g
(
x
) = 1
/x
2
.
3. The upper range of integration is
x
= 2.
4. The lower range of integration is the
x
value where the line
y
=
x
intersects the curve
y
= 1
/x
2
. That is, we set the expressions for the
y
values equal to each other and solve for
x
:
y
=
y
x
=
1
x
2
x
3
= 1
x
= 1
.
The lower range of integration is
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
 Fall '10
 KIHYUNHYUN
 Calculus

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