is a cylinder, with radius a, in three dimensions. It can be written in parametric equations using cylindrical coordinates but taking r= a: , , .

The "position vector" of any point on the cylinder can be written as . The two derivatives:

and

are tangent vectors to the cylinder. Their cross product is

and the "scalar differential of surface area" is given by the length of that: .

(For this simple geometry, you could have just noted that the differential of lengtharoundthe cylinder is so the differential of area is .)

Since this integral is

.

(In fact, just thinking about the "symmetry" (or, more correctly "asymmetry") makes this problem trivial.)