When you have a surface defined in terms of two parameters, x= f(u,v), y= g(u,v), z= h(u,v), you can write each point as the vector equation, . The two partial derivatives, with respect to u and v are and . The "fundamental vector product" for the surface is .

Finally, the "differential of surface area is the magnitude of that product time drdt:

Here, since this is rotated around the x-axis, x= t, as you say, and so r(t), perpendicular to the x, axis, is . , , and . .