Math 152 – Spring 2016
Section 9.4
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Section 9.4 – Area of a Surface of Revolution
Define ds to be the inside of the integral from the arc length formula. There were three
possibilites:
s
2
dy
dx
ds = 1 +
dx
s 2
dx
ds =
+ 1 dy
dy
s 2
2
dx
dy
+
dt
ds =
dt
dt
Suppose that we rotate a curve around either the x- or y-axis to create a solid. We
want to find the surface area of the solid we have created.
Area of a Surface of Revolution The formula for the area of a surface of
revolution is
Z
S=
b
2πrds
a
where r is the radius and ds is one of the three choices above. There are three decisions
to make for this formula:
•
Radius.
If the function is rotated around the x-axis, then the radius is y.
If the function is rotated around the y-axis, then the radius if x.
•
ds
r
2
dy
1 + dx
dx.
r 2
dx
+ 1 dy.
If x = g(y), choose ds =
dy
r
If y = f (x), choose ds =
If x = f (t) and y = g(t), choose ds =
•
dx 2
dt
+
dy
dt
2
dt.
Variable
Choose the limits based on which variable is being integrated (dx, dy, dt).
Make sure the radius, x or y, matches the variable being integrated dx, dy, dt. If
they do not match, use the given function to replace the radius. For example, in
a dx integral with radius y, replace y with f (x). In a dx interval with radius x,
leave the x alone.
Math 152 – Spring 2016
Section 9.4
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Example 1. Find the area of the surface obtained by rotating the curve about the
given axis.
(a) y = sin x, 0 ≤ x ≤ π, x-axis
(b) 2y + x2 = 1, 0 ≤ x ≤ 1, y-axis
(c) x − 1 = 2y 2 , 1 ≤ y ≤ 2, x-axis
Math 152 – Spring 2016
(d) x =
Section 9.4
p
2y − y 2 , 0 ≤ y ≤ 1, y-axis
(e) x = et − t, y = 4et/2 , 0 ≤ t ≤ 1 y-axis
Example 2. Show that the surface area of a sphere with radius r is 4πr2 .
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Math 152 – Spring 2016
Section 9.4
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Formulas for Area of a Surface of Revolution
y = f (x)
S=
rotate about x-axis
rotate about y-axis
(radius = y)
(radius = x)
Rb
a
x = g(y)
S=
x = f (t), y = g(t)
Rb
S=
a
r
2πf (x) 1 +
Rb
a
2πy
r
r
2πg(t)
dx
dy
2
dx 2
dt
dy
dx
2
dx
+ 1 dy
+
dy
dt
2
S=
S=
dt
S=
Rb
a
Rb
Rb
a
a
r
2πx 1 +
2πg(y)
r
r
2πf (t)
dx
dy
dx 2
dt
dy
dx
2
+
2
dx
+ 1 dy
dy
dt
2
dt