  In Microscopic Particle Image Velocimetry ($\mu$PIV), velocity fields in
microchannels are sampled over finite volumes within which the velocity fields
themselves may vary significantly. In the past, this has limited measurements
often to be only qualitative in nature, blind to velocity magnitudes. In the
pursuit of quantitatively useful results, one has treated the effects of the
finite volume as errors that must be corrected by means of ever more
complicated processing techniques. Resulting measurements have limited
robustness and require convoluted efforts to understand measurement
uncertainties. To increase the simplicity and utility of $\mu$PIV measurements,
we introduce a straightforward method, based directly on measurement, by which
one can determine the size and shape of the volume over which moving fluids are
sampled. By comparing measurements with simulation, we verify that this method
enables quantitative measurement of velocity profiles across entire channels,
as well as an understanding of experimental uncertainties. We show how the
method permits measurement of an unknown flow rate through a channel of known
geometry. We demonstrate the method to be robust against common sources of
experimental uncertainty. We also apply the theory to model the technique of
Scanning $\mu$PIV, which is often used to locate the center of a channel, and
we show how and why it can in fact misidentify the center. The results have
general implications for research and development that requires reliable,
quantitative measurement of fluid flow on the micrometer scale and below.
