Section 9.4 Surface Area of a Surface of Revolution
If a curve C is rotated about the x-axis or about the y-axis the surface area of the resulting surface is
b
S  2   r dl where dl can be any of the three arc length forms depending on the description of the
a
curve.
If C is given parametrically as
t1
x  x (t )
y  y (t )
t 0  t  t1
S  2   r ( t ) ( x ' ( t ))  ( y ' ( t )) dt where r(t) is x(t) if C is rotated about the y-axis and
2
2
t0
r(t) is y(t) if C is rotated about the x-axis.
If C is given as y = f(x), let t=x and replace x ‘(t) with 1.
If C is given as x = g(y), let t=y and replace y’(t) with 1.
Examples
1. Show that the surface area of a sphere of radius r is S  4  r .
2
2. The curve y  e
x
0  x  1 is rotated about the x-axis.
3. Set up only if the curve y  e
x
0  x  1 is rotated about the y-axis.
4. Find the surface area of the torus (doughnut) formed if the circle ( x  R )  y
2
Parameterize the curve as
x ( t )  R  r cos t
y ( t )  r sin t
2
r .
0  t  2
2