Section 5.1: Areas and Distances
Instructor: Ms. Hoa Nguyen
([email protected])
The Area Problem
Find the area of the region
S
that lies under the curve
y
=
f
(
x
) from
a
to
b
. This means that
S
is bounded
by the graph of a continuous function
f
(where
f
(
x
)
≥
0), the vertical lines
x
=
a
and
x
=
b
, and the xaxis.
Example 1
: Find the area under the parabola
y
=
x
2
from 0 to 1.
First, we approximate the region
S
by rectangles and then we take the limit of the areas of these rectangles
as we increase the number of rectangles.
Divide
S
into 4 strips, and approximate each strip by a rectangle. The height of each rectange is the value
of
f
at the RIGHT endpoint of each subinterval.
The sum of the areas of these four rectangles is:
R
4
=
1
4
·
(
1
4
)
2
+
1
4
·
(
1
2
)
2
+
1
4
·
(
3
4
)
2
+
1
4
·
(1)
2
=
15
32
= 0
.
46875
Instead of using the above rectangles, we use the smaller rectangles. Divide
S
into 4 strips, and approximate
each strip by a rectangle. The height of each rectange is the value of
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 Fall '08
 Noohi
 Calculus, Continuous function, Rectangle

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