gutierrez (ig3472) – HW05 – Neitzke – (56585)
1
This
printout
should
have
22
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
001
10.0 points
Determine the volume of the right circular
cone generated by rotating the line
x
=
3
4
y
about the
y
axis between
y
= 0 and
y
= 4.
1.
V
= 10
π
cu.units
2.
V
= 13
π
cu.units
3.
V
= 12
π
cu.units
correct
4.
V
= 9
π
cu.units
5.
V
= 11
π
cu.units
Explanation:
The volume,
V
, of the solid of revolution
generated by rotating the graph of
x
=
f
(
y
)
about the
y
axis between
y
=
a
and
y
=
b
is
given by
V
=
π
integraldisplay
b
a
f
(
y
)
2
dy.
When
f
(
y
) =
3
4
y
and
a
= 0
, b
= 4, therefore,
V
=
π
integraldisplay
b
a
9
16
y
2
dx
=
π
bracketleftBig
3
16
y
3
bracketrightBig
4
0
.
Consequently,
V
= 12
π
cu.units
.
002
10.0 points
Find the volume of the paraboloid gener
ated by rotating the graph of
y
= 6
√
x
be
tween
x
= 0 and
x
= 2 about the
x
axis.
1.
volume = 70
π
cu.units
2.
volume = 71
π
cu.units
3.
volume = 68
π
cu.units
4.
volume = 72
π
cu.units
correct
5.
volume = 69
π
cu.units
Explanation:
The solid of revolution generated by rotat
ing the graph of
y
=
f
(
x
) about the
x
axis
between
x
=
a
and
x
=
b
has
volume =
π
integraldisplay
b
a
f
(
x
)
2
dx .
When
f
(
x
) = 6
√
x,
a
= 0
,
b
= 2
,
therefore,
V
=
π
integraldisplay
2
0
36
x dx
=
π
2
bracketleftBig
36
x
2
bracketrightBig
2
0
.
Consequently,
V
= 72
π
cu.units
.
keywords:
volume, integral, solid of revolu
tion
003
10.0 points
Find the volume,
V
, of the solid obtained
by rotating the region bounded by
y
=
x
2
,
x
= 0
,
y
= 4
about the
y
axis. (Hint:
as always graph the
region first
).
1.
V
= 4 cu. units
2.
V
=
16
3
cu. units
3.
V
=
16
3
π
cu. units
4.
V
= 8 cu. units
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
gutierrez (ig3472) – HW05 – Neitzke – (56585)
2
5.
V
= 4
π
cu. units
6.
V
= 8
π
cu. units
correct
Explanation:
The region rotated about the
y
axis is sim
ilar to the shaded region in
4
y
x
(not drawn to scale). Now the volume of the
solid of revolution generated by revolving the
graph of
x
=
f
(
y
) on the interval [
a, b
]
on the
y
axis
about the
y
axis is given by
volume =
π
integraldisplay
b
a
f
(
y
)
2
dy .
To apply this we have first to express
x
as
a function of
y
since initially
y
is defined in
terms of
x
by
y
=
x
2
. But after taking square
roots we see that
x
=
y
1
/
2
. Thus
V
=
π
integraldisplay
4
0
y dy
=
π
bracketleftbigg
1
2
y
2
bracketrightbigg
4
0
.
Consequently,
V
= 8
π
.
004
10.0 points
Let
A
be the bounded region enclosed by
the graphs of
f
(
x
) =
x ,
g
(
x
) =
x
4
.
Find the volume of the solid obtained by ro
tating the region
A
about the line
x
+ 4 = 0
.
1.
volume =
11
15
π
2.
volume =
71
15
π
3.
volume =
41
15
π
correct
4.
volume =
26
15
π
5.
volume =
56
15
π
Explanation:
The solid is obtained by rotating the shaded
region about the line
x
+ 4 = 0 as shown in
1
x
+ 4 = 0
(not drawn to scale). To compute the volume
of this solid we use the washer method.
For
this we have to express
f
and
g
as functions
of
y
:
y
=
f
(
x
) =
x
=
⇒
x
=
y ,
while
y
=
g
(
x
) =
x
4
=
⇒
x
=
y
1
/
4
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 GOGOLEV
 Calculus, dx, Gutierrez

Click to edit the document details