let V be the unit ball in three space centered at the origin: 0≦r≦1, find
how do I calculate this integration with vector calculus knowledge?
I think I need to use spherical polar coordinates, but it makes me confuse.
No, it is not correct- what you wrote makes no sense. You cannot have a double integral over a three dimensional body. You need to integrate either over the unit ball or over the unit sphere (the surface of the unit ball).
In either case, because the region is symmetric while the integrand, x, is an odd function, the integral is 0.