The sin(theta) component is the y-contribution of the particulars and the cos(theta) component is the x-component.
Think of the projection of each slice of the cylinder: you are adding up each individual slice (i.e. every cross section of the flattened tube) and this is weighted by the density of molecules per cross-sectional area (which is assumed constant) and then you add these up in the limit of the integral.
Now each slice is pi*r^2, but the length of all the slices will be ct*cos(theta) and the density of each individual slice will be density*sin(theta) of which the density is considered constant as (N/V) throughout the whole cylinder.
So we are calculating individual slice density * slice volume (infinitesimal) * length of cylinder or in other words, calculate each individual slice-density in first two terms and add up all these contributions. The density of each slice will be the total density * sin(theta) (just a projection of a vector running along the line ct projected to the the centre of that cylinder).
I don't know where the surface of the area comes in: maybe you could explain that argument but with regards to the other part, I've given my interpretation. Also I'm not sure why theta is being integrated with respect to (since it should be fixed) but the other one makes sense.
So in short (if it's not completely right, maybe you can build on it to make it right) you have a projection in both sin(theta) and cos(theta) where sin(theta) provides the projection of the density of a particular slice and cos(theta) provides the total length of the deformed tube and when slice volume * molecule density * number of slices through integral is computed you get total number of molecules travelling in that direction.