Since the density varies from 1 slug/ft^{3} at the base to 2
slug/ft^{3} at the tip, the density is given by the equation

ρ(x) = 1 + 0.01x
slug/ft^{3}

The density does not vary in the y or z direction, therefore

Substitute the density and area equations into the center of mass equation
gives

Expand and integrate gives

The final result is

Notice the distance is slightly larger than the centroid, which corresponds
to the increasing density toward the front.

Center of Gravity

Since the nose cone is relatively small, variation of gravity
over its volume can be neglected. Thus, the center of gravity
and the center of mass
are the same point.