I think you need to find a parameterisation of the strip X(phi, r), calculate the cross product of the unit vectors (dX/dphi cross dX/dr), then integrate that (

I think you need to find a parameterisation of the strip X(phi, r), calculate the cross product of the unit vectors (dX/dphi cross dX/dr), then integrate that (X being a 3D vector while phi, r are scalar parameters).