If anyone could explain how this problem is done, it would be very much appreciated!
Write the equation for the surface generated by revolving the given around the indicated axis.
4x + 9y^2 = 36 around the y-axis
The solid (I can't think of a name for this thing) has, as a cross-section on the xy plane a pair of parabolas both with vertices at the origin and axes of symmetry along the +x and -x axes. As we move along the y direction the cross-sections in the xz plane will be circular, so we need the equation of a circle in the xz plane centered on the origin. That would be
$\displaystyle x^2 + z^2 = r^2$
where r is the radius of the circle in the cross-section, which will depend on the value of y we choose for the cross-section. The radius will be equal to the distance from the point on the surface to the y-axis, which in this case will be measured by the x value. Thus the radius is given by the original equation:
$\displaystyle r = x = -\frac{9}{4}y^2 + 9$
and the surface becomes:
$\displaystyle x^2 + z^2 = \left ( -\frac{9}{4}y^2 + 9 \right )^2$
or
$\displaystyle x^2 - \left ( \frac{9}{4}y^2 - 9 \right )^2 + z^2 = 0$
-Dan