momin (rrm497) – Homework 02 – cheng – (58520)
1
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beFore answering.
001
10.0 points
Stewart Section 5.1, Example 3(b), page 321
Estimate the area,
A
,under the graph oF
f
(
x
) = 2 sin
x
between
x
= 0 and
x
=
π
3
using fve approx
imating rectangles oF equal widths and right
endpoints.
1.
A
≈
1
.
178
correct
2.
A
≈
1
.
138
3.
A
≈
1
.
098
4.
A
≈
1
.
118
5.
A
≈
1
.
158
Explanation:
An estimate For the area,
A
, under the
graph oF
f
on [0
, b
] with [0
, b
] partitioned in
n
equal subintervals
[
x
i

1
, x
i
] =
b
(
i

1)
b
n
,
ib
n
B
and right endpoints
x
i
as sample points is
A
≈
±
f
(
x
1
) +
f
(
x
2
) +
. . .
+
f
(
x
n
)
²
b
n
.
±or the given area,
f
(
x
) = 2 sin
x,
b
=
π
3
,
n
= 5
,
and
x
1
=
1
15
π,
x
2
=
2
15
π,
x
3
=
1
5
π,
x
4
=
4
15
π,
x
5
=
1
3
π .
Thus
A
≈
2
±
sin(
1
15
π
) +
. . .
+ sin(
1
3
π
)
²
π
15
.
AFter calculating these values we obtain the
estimate
A
≈
1
.
178
For the area under the graph.
002
10.0 points
Rewrite the sum
±
3+
p
1
9
P
2
²
+
±
6+
p
2
9
P
2
²
+
. . .
+
±
24+
p
8
9
P
2
²
using sigma notation.
1.
9
s
i
= 1
±
3
i
+
p
i
9
P
2
²
2.
9
s
i
= 1
3
±
i
+
p
3
i
9
P
2
²
3.
9
s
i
= 1
3
±
i
+
p
i
9
P
2
²
4.
8
s
i
= 1
±
i
+
p
3
i
9
P
2
²
5.
8
s
i
= 1
3
±
i
+
p
i
9
P
2
²
6.
8
s
i
= 1
±
3
i
+
p
i
9
P
2
²
correct
Explanation:
The terms are oF the Form
±
3
i
+
p
i
9
P
2
²
,
with
i
= 1
,
2
, . . . ,
8. Consequently, in sigma
notation the sum becomes
8
s
i
= 1
±
3
i
+
p
i
9
P
2
²
.
003
10.0 points