Surface of revolution: equation, points, area

A surface of revolution is formed by rotating a plane curve (the generatrix or profile) about an axis in its plane. Every point of the curve moves on a circle (parallel); each position of the curve is a meridian. For a profile $r=f(s)$, $z=g(s)$ about the Z axis:

$$\mathbf P(s,\phi)=\big(f(s)\cos\phi,\ f(s)\sin\phi,\ g(s)\big),\qquad 0\le\phi\le2\pi$$

Generatrix

The line is $r=2+2s$, $z=6s$ for $0\le s\le1$.

$$\mathbf P(s,\phi)=\big((2+2s)\cos\phi,\ (2+2s)\sin\phi,\ 6s\big)$$

The surface is the side of a frustum of a cone with bottom radius 2, top radius 4 and height 6.

Points

Curved surface area

Slant length $l=\sqrt{(4-2)^2+6^2}=\sqrt{40}=6.325$.

$$A=\pi\,(r_1+r_2)\,l=\pi(2+4)(6.325)=119.2\ \text{square units}$$

Answer: P(s, phi) = ((2+2s)cos phi, (2+2s)sin phi, 6s); points (0,3,3), (-4,0,6), (1.768,1.768,1.5); area = 119.2 square units.