No-- you kind of missed the point

alpha is the curve of intersection of the plane and the cylinder

See the attachment for the details where the line integral is computed directly

If you use Stokes theorem then your surface is z= 1-x-y over a circle of radius 1

N = i + j + k

curl F*N = 3 int(curlF*NdA) = 3*area = 3 pi which is what we obtain computing the line integral directly