AREAS AND DISTANCES
Suppose that we want to find the area under a curve. First of all, we need to
define what the area is. If we have a rectangle, it is relatively easy, because we can
simply define the area as the product of the length and the width. We will use this
property of a rectangle as a base point of defining the area under a curve.
Example 1.
Use rectangles to estimate the area under the parabola
y
=
x
2
from 0 to 1.
Solution
Suppose we divide the parabolic region into four strips.
We can ap
proximate each strip by a rectangle whose base is the same as the strip and whose
height is the same as the right edge of the strip.
Each rectangle has width
1
4
and the heights are (
1
4
)
2
, (
2
4
)
2
, (
3
4
)
2
, and 1
2
. If we
let
R
4
be the sum of the areas of these rectangles, we get
R
4
=
1
4
(
1
4
)
2
+
1
4
(
2
4
)
2
+
1
4
(
3
4
)
2
+
1
4
·
1
2
=
15
32
= 0
.
46875
We can also see that the area
A
of the parabolic region is less than
R
4
, so
A <
0
.
46875
Alternatively, we can use smaller rectangle whose heights are the values of
f
at
the endpoints of the subintervals. The sum of these approximating rectangles is
L
4
=
1
4
·
0
2
+
1
4
(
1
4
)
2
+
1
4
(
2
4
)
2
+
1
4
(
3
4
)
2
=
7
32
= 0
.
21875
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 Fall '09
 BenedictGross
 Rectangle

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