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EXERCISES 9.9
±ind the length of the curve, locate the centroid, and determine
the area of the surface generated by revolving the curve about
the
x
axis.
1.
f
(
x
)
=
4,
x
∈
[0, 1].
2.
f
(
x
)
=
2
x
,
x
∈
[0, 1].
3.
y
=
4
3
x
,
x
∈
[0, 3].
4.
y
=−
12
5
x
+
12,
x
∈
[0, 5].
5.
x
(
t
)
=
3
t
,
y
(
t
)
=
4
t
;
t
∈
[0, 2].
6.
r
=
5,
θ
∈
[0,
1
4
π
].
7.
x
(
t
)
=
2 cos
t
,
y
(
t
)
=
2 sin
t
;
t
∈
[0,
1
6
π
].
8.
x
(
t
)
=
cos
3
t
,
y
(
t
)
=
sin
3
t
;
t
∈
[0,
1
2
π
].
9.
x
2
+
y
2
=
a
2
,
x
∈
[
−
1
2
a
,
1
2
a
],
y
>
0,
a
>
0.
10.
r
=
1
+
cos
θ
,
θ
∈
[0,
π
].
±ind the area of the surface generated by revolving the curve
about the
x
axis.
11.
f
(
x
)
=
1
3
x
3
,
x
∈
[0, 2].
12.
f
(
x
)
=
√
x
,
x
∈
[1, 2].
13.
4
y
=
x
3
,
x
∈
[0, 1].
14.
y
2
=
9
x
,
x
∈
[0, 4].
15.
y
=
cos
x
,
x
∈
[0,
1
2
π
].
16.
f
(
x
)
=
2
√
1
−
x
,
x
∈
[
−
1, 0].
17.
r
=
e
θ
,
θ
∈
[0,
1
2
π
].
18.
y
=
cosh
x
,
x
∈
[0, ln 2].
19.
Take a > 0. One arch of a cycloid can be deFned parametri
cally by setting
x
(
θ
)
=
a
(
θ
−
sin
θ
),
y
(
θ
)
=
a
(1
−
cos
θ
),
Letting
θ
range from 0 to 2
π
.
(a) ±ind the area under the curve.
(b) ±ind the area of the surface generated by revolving the
arch about the
x
axis.
±
c
20.
Take
a
>
0. The curve
x
(
θ
)
=
3
a
cos
θ
+
a
cos 3
θ
,
y
(
θ
)
=
3
a
sin
θ
−
a
sin 3
θ
,
with
θ
ranging from 0 to 2
π
, is called a
hypocycloid
.
(a) Use a graphing utility to draw the curve.
(b) ±ind the area enclosed by the curve.
(c) Set up a deFnite integral that gives the area of the surface
generated by revolving the curve about the
x
axis.
21.
By cutting a cone of slant height
s
and base radius
r
along
a lateral edge and laying the surface ²at, we can form a
sector of a circle of radius
s
. (See the Fgure.) Use this idea
to verify ±ormula (9.9.1).
r
2
r
π
s
s
,
in radians
area =
1
2
2
θ
s
22.
The Fgure shows a ring formed by two quartercircles. Call
the corresponding quarterdiscs
±
a
and
±
r
. By Section
6.4,
±
a
has centroid (4
a
/
3
π
,4
a
/
3
π
) and
±
r
has centroid
(4
r
/
3
π
r
/
3
π
).
y
x
a
r
r
a
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±
CHAPTER 9
THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS
(a) Without integration, calculate the centroid of the ring.
(b) Find the centroid of the outer arc from your answer to
part (a) by letting
a
tend to
r
.
23.
(a) Find the centroid of each side of the triangle in the ±gure.
y
x
5
4
3
(b) Use your answers in part (a) to calculate the centroid of
the triangle.
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston  Downtown.
 Spring '10
 SMITH
 Calculus, Formulas

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