I am trying to convert the volume integral of the cross-product between position and velocity vector into surface integral.

The assumption is that the velocity field is solenoidal (i.e. divergence of the velocity field is zero) and is two-dimensional.

I tried an approach as shown in the picture below but it still gives me the volume integral term and I cannot get rid of it using the given assumptions.

Is there any other approach that I can try or is it not possible to convert this volume integral to a surface integral?