Math 152 Class Notes
October 15, 2015
9.4 Area of a Surface of Revolution
In this section, we study the area of a surface of revolution. Suppose a curve from
to
Q
is rotated about the
x-axis.
P
We obtain a surface of revolution.
The surface area of the resulting surface is given by
ˆ
Q
S=
2πy ds
P
According to the form of curve equation, we have various explicit formulas as follows.
(a)
y = f (x), a ≤ x ≤ b
If the curve is given by
and rotated about the
x-axis,
then
x-axis,
then
the resulting surface area is given by
ˆ
s
b
S=
1+
2πf (x)
a
(b)
If the curve is given by
x = g(y), c ≤ y ≤ d
dy
dx
2
dx
and rotated about the
the resulting surface area is given by
ˆ
S=
s
d
2πy
1+
c
(c)
If the curve is given by
x-axis,
dx
dy
2
dy
x = f (t), y = g(t), α ≤ t ≤ β
then the resulting surface area is given by
ˆ
S=
s
β
2πg(t)
α
dx
dt
2
+
dy
dt
2
dt
and rotated about the
When a curve from
P
to
Q is rotated about the y -axis, the surface area of the resulting
surface is given by
ˆ
Q
S=
2πx ds
P
As above, we have the following explicit formulas corresponding to various forms of
the curve equation.
(a)
y = f (x), a ≤ x ≤ b
If the curve is given by
and rotated about the
y -axis,
then
y -axis,
then
the resulting surface area is given by
ˆ
S=
s
b
2πx
1+
a
(b)
dy
dx
x = g(y), c ≤ y ≤ d
If the curve is given by
2
dx
and rotated about the
the resulting surface area is given by
ˆ
S=
s
d
2πg(y)
1+
c
(c)
If the curve is given by
y -axis,
dx
dy
2
dy
x = f (t), y = g(t), α ≤ t ≤ β
then the resulting surface area is given by
ˆ
s
β
2πf (t)
S=
α
dx
dt
2
+
dy
dt
2
dt
and rotated about the
In practice, all the formulas can be summarized and remembered as
ˆ
Q
S=
2πr ds
P
where
ds
is the arc length and
r
is the radius, i.e. the distance from the curve to the
axis of revolution.
Example 1. Find the surface area obtained by rotating the curve
x≤2
about the
x-axis.
y=
√
4 − x2 , −2 ≤
Example 2. Find the surface area obtained by rotating the curve
about the
y -axis.
x2
y= ,0≤x≤1
2
Example 3. Find the surface area obtained by rotating the curve
0≤t≤1
about the
x-axis.
Example 4. Find the surface area obtained by rotating the curve
0≤t≤1
about the
x = 3t − t3 , y = 3t2 ,
y -axis.
x = et − t, y = 4et/2 ,