Section 15.7: Triple Integration in Cylindrical Coordinates

SOLUTION

The equation says that the z-value, or height, of each point on the surface is the same as r, the distance from the point to the z-axis. Because \(\theta\) doesn’t appear, it can vary. So any horizontal trace in the plane \(z=k\) (\(k>0\)) is a circle of radius k. These traces suggest that the surface is a cone. This prediction can be confirmed by converting the equation into rectangular coordinates. From the first equation in (2) we have \[ z^2 = r^2 = x^2 + y^2 \] We recognize the equation \(z^2 = x^2 + y^2\) (by comparison with Table 1 in Section 12.6) as being a circular cone whose axis is the z-axis (see Figure 5).