  To determine the magnification of an extended source caused by gravitational
lensing one has to perform a two-dimensional integral over point-source
magnifications in general. Since the point-source magnification jumps to an
infinite value on caustics, special care is required. For a uniformly bright
source, it has been shown earlier that the calculation simplifies if one
determines the magnification from the area of the images of the extended source
by applying Green's theorem so that one ends up with a one-dimensional
integration over the image boundaries. This approach is discussed here in
detail, and it is shown that it can be used to yield a robust and efficient
method also for limb-darkened sources. It is also shown that the centroid shift
can be calculated in a similar way.
