Begin2.DVI

axis. A paraboloid is obtained by rotating the parabola y=x^2 , 0 ≤x≤x 0 about the

x= 0 axis.

The general procedure for determining the equation for representing a surface of

revolution is as follows. First select a general point P on the given curve and then

rotate the point Pabout the axis of revolution to form a circle. This usually involves

some parameter used to describe the general point. One can then determine the

equation of the surface by eliminating the parameter from the resulting equations.

Example 7-7. A curve y=f(x) for a≤x≤bis rotated about the x−axis.

Find the equation describing the surface of revolution.

Solution A general point P on the given

curve, when rotated about the x−axis pro-

duces the circle y^2 +z^2 =r^2 , where r=f(x)

is the radius of the circle. Eliminating the

parameter rgives the equation of the surface

of revolution as y^2 +z^2 = [f(x)]^2

Example 7-8. The curve y=f(z)for a≤z≤bis rotated about the z−axis.

Find the equation describing the surface of revolution.

Solution A general point P on the given

curve is rotated about the z−axis to form

the circle x^2 +y^2 =r^2 where r=f(z)is the

radius of the circle. Eliminating rbetween

these two equations gives the equation for

the surface of revolution as x^2 +y^2 = [f(z)]^2

Example 7-9. A curve described by the parametric equations x=x(t),y=y(t),

z=z(t)for t 0 ≤t≤t 1 , is rotated about the line

x−x 0

b 1

=y−y^0

b 2

=z−z^0

b 3

where
b=b 1 eˆ 1 +b 2 ˆe 2 +b 3 ˆe 3 is the direction vector of the line. Find the equation of

the surface of revolution.