saliyev (is4663) – Quest Assignment 2 – rodin – (54520)
1
This printout should have 10 questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
001
10.0 points
Oil is ±owing From a well in a continuous
stream at a rate oF
f
(
t
)=
14000
(
t
+5)
2
barrels per month. ²ind the total oil produced
by the well during its liFetime operation.
1.
#barre
ls =3200
2.
ls =3000
3.
ls =2800
correct
4.
ls =2400
5.
ls =2600
Explanation:
The liFetime production oF oil is given by
the improper integral
I
=
±
∞
0
f
(
t
)
dt
=
±
∞
0
14000
(
t
2
dt.
As a limit
I
=l
im
s
→∞
±
s
0
14000
(
t
2
dt
oF proper integrals, thereFore,
I
s
²

14000
(
t
+5
³
s
0
=2800
.
Consequently,
ls = 2800
.
002
10.0 points
²ind the area oF the region
R
defned by
R
=
´
(
x, y
):
x
≥
4
,
0
≤
y
≤
e

x/
7
µ
.
1.
Area(
R
)=7
e

4
/
7
2.
Area(
R
e
4
/
7
3.
Area(
R
e
4
/
7
correct
4.
Area(
R
e

4
/
7
5.
Area(
R
∞
Explanation:
The region
R
is the area below the graph
oF
y
=
e

x/
7
on the infnite interval [

4
,
∞
),
and so looks like the shaded region in
x
y

4
extended to +
∞
in the
x
direction. Thus the
area oF
R
is the limit
lim
t
±
t

4
e

x/
7
dx .
Now
±
t

4
e

x/
7
dx
=

7
²
e

x/
7
³
t

4
=

7
e

t/
7
+7
e
4
/
7
.
Consequently,
Area(
R
e
4
/
7
since
lim
x
e

x
=0
.
003
10.0 points
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2
Which (if any) of the following integrals is
convergent?
1.
±
∞

5
6
x
2
dx
2.
±
∞
5
3
x
√
xdx
3.
±
5
0
4
√
x
dx
correct
4.
none of them
5.
±
5
0
8
x
√
x
dx
6.
±
0

5
6
x
2
dx
Explanation:
Each of the improper integrals
I
=
±
5
0
1
x
p
dx,
I
=
±
0

5
1
x
p
dx
is convergent if and only if
p<
1, while each
of
I
=
±
∞
5
1
x
p
dx,
I
=
±
∞

5
1
x
p
dx
is convergent if and only if
p>
1. Of the given
integrals, therefore, only
I
=
±
5
0
4
√
x
dx
is convergent.
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 Spring '11
 RUSIN

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