X-ray fluorescence computed tomography (XFCT) is a synchrotron-based imaging modality employed for mapping the distribution of elements within slices or volumes of intact specimens. A pencil beam of external radiation is used to stimulate emission of characteristic X-rays from within a sample, which is scanned and rotated through the pencil beam in a first-generation tomographic geometry. One limitation of XFCT is the long image acquisition time required to acquire a complete set of line integrals one-by-one. Typically, even if only a portion of a slice through the object is of interest, measurement lines are acquired spanning the entire object at every projection view over 180 degrees to avoid reconstructing images with so-called truncation artifacts. In this work, we show that when attenuation is negligible, recent developments in tomographic reconstruction theory can be used to reduce the scanning effort required to reconstruct regions of interest within the slice. The new theory provides explicit guidance as to which line integrals must be measured for a given ROI and also provides a backprojection-filtration reconstruction algorithm that averts the truncation artifacts that typically plague filtered backprojection reconstructions from truncated data. This is demonstrated through simulation studies and with real synchrotron-based XFCT data.