  In a line caustic crossing microlensing event, the caustic line moving across
the surface of the source star provides a direct method to measure the
integrated luminosity profile of the star. Combined with the enormous
brightening at the caustic crossings, microlensing offers a promising tool for
studying stellar luminosity profiles. We derive the amplification behavior of
the two extra images that become partial images conjoined across the critical
curve at a line caustic crossing. We identify the multiplicative factors that
depend on the caustic crossing point and the relative size of the star, and the
shape function that depends on the stellar luminosity profile. We examine the
analytic limb-darkening models -- linear, square root, and square -- using the
analytic form of the shape function. We find that the microlensing lightcurves
must be determined to an accuracy of better than 0.3-0.8% in order to be able
to determine the linear limb-darkening parameter $c_1$ with precision of
$\delta c_1 = 0.1$. This is similar to the accuracy level required in eclipsing
binaries as reported by Popper (1984).
