Math 20B
Final Exam Solutions
Winter 2001
1. (40 pts.) Let
R
be the region between the two parabolas
y
=
x
2
and
x
=
y
2
.
Let
V
be the volume obtained when
R
is rotated about the
y
axis.
(a) Sketch the region
R
. Include in your drawing the coordinates of the point where the parabolas
intersect.
Ans.
We omit the picture. The region lies in the first quadrant, is bounded below by
y
=
x
2
and
above by
x
=
y
2
, and the points of intersection are (0,0) and (1,1). You can find the intersection
points from your sketch (and then easily check them in your head). Alternatively, you can find
them by solving the equations: Squaring the first and using the second to eliminate
y
gives
x
4
=
x
and so either
x
= 0 or
x
3
= 1. The solution to the latter is
x
= 1. From
y
=
x
2
, we
find the corresponding values of
y
.
(b) The arc length of the boundary of
R
that is on the parabola
y
=
x
2
.
Ans.
Z
1
0
p
1 + 4
x
2
dx
or
Z
1
0
p
1 + 4
y
2
dy
(c) The volume of
V
.
Ans.
Z
1
0
π
‡
√
y
2

(
y
2
)
2
·
dy
=
Z
1
0
π
(
y

y
4
)
dy
(d) The surface area of
V
.
Be careful
:
V
has what might be called an inner and outer surface.
The surface area is the sum of the areas of these two surfaces.
Ans.
Z
1
0
2
π
√
y
q
1 + (1
/
2
y
1
/
2
)
2
dy
+
Z
1
0
2
πy
2
p
1 + (2
y
)
2
dy
=
Z
1
0
2
π
p
y
+ 1
/
4
dy
+
Z
1
0
2
πy
2
p
1 + 4
y
2
dy
2. (20 pts.) Given the two curves
r
= 2 and
r
= 4 cos
θ
in polar coordinates.
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 Winter '08
 Justin
 Math, Calculus, 10 pts, 20 pts, 15 pts, 40 pts, 30 pts

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