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Math 21B
UC Davis, Winter 2010
Dan Romik
Solutions to practice questions for midterm exam 2
1.
Let
C
denote the curve
y
= 2
 
x

,

1
≤
x
≤
1.
(a) Compute the volume of the solid of revolution formed by revolving the region
bounded between
C
and the
x
axis about the
x
axis.
(b) Compute the surface area of the surface of revolution of
C
about the
x
axis.
(c) Compute the arc length of
C
.
Solution.
In all three cases, we can use symmetry by computing the answer for the
part of the curve in the range 0
≤
x
≤
1 and multiplying it by 2. This gives
(a)
V
=
Z
1

1
πy
2
dx
= 2
Z
1
0
πy
2
dx
= 2
π
Z
1
0
(2

x
)
2
dx
= 2
π

(2

x
)
3
3
±
±
±
1
0
=
2
π
3
(2
3

1
3
) =
14
π
3
,
(c)
L
= 2
Z
1
0
s
1 +
²
dy
dx
³
2
dx
= 2
Z
1
0
√
2
dx
= 2
√
2
,
(b)
A
= 2
Z
1
0
2
π
(2

x
)
√
2
dx
=
...
= 6
√
2
π.
2.
The parametric curve
x
=
2
3
t
3
/
2
,
y
= 2
√
t,
0
≤
t
≤
√
3
is revolved about the
y
axis. Find the area of the resulting surface.
1
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View Full DocumentSolution.
The diﬀerential arclength is given by
ds
=
p
dx
2
+
dy
2
=
s
±
dx
dt
²
2
+
±
dy
dt
²
2
dt
=
√
t
+
t

1
dt
=
r
t
2
+ 1
t
dt,
so the surface area is
A
=
Z
√
3
0
2
πx
(
t
)
r
t
2
+ 1
t
dt
=
Z
√
3
0
2
π
·
2
3
·
t
√
t
2
+ 1
dt
This integral can be computed by making the substitution
u
=
t
2
+ 1, leading to
A
=
2
π
3
Z
4
1
√
udu
=
...
=
28
π
9
.
3.
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 Winter '08
 Vershynin
 Math, Differential Equations, Equations

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