We present scaling relations between the integrated Sunyaev-Zeldovich Effect (SZE) signal, $Y_{rm SZ}$, its X-ray analogue, $Y_{rm X}equiv M_{rm gas}T_{rm X}$, and total mass, $M_{rm tot}$, for the 45 galaxy clusters in the Bolocam X-ray-SZ (BOXSZ) sample. All parameters are integrated within $r_{2500}$. $Y_{2500}$ values are measured using SZE data collected with Bolocam, operating at 140 GHz at the Caltech Submillimeter Observatory (CSO). The temperature, $T_{rm X}$, and mass, $M_{rm gas,2500}$, of the intracluster medium are determined using X-ray data collected with Chandra, and $M_{rm tot}$ is derived from $M_{rm gas}$ assuming a constant gas mass fraction. Our analysis accounts for several potential sources of bias, including: selection effects, contamination from radio point sources, and the loss of SZE signal due to noise filtering and beam-smoothing effects. We measure the $Y_{2500}$--$Y_{rm X}$ scaling to have a power-law index of $0.84pm0.07$, and a fractional intrinsic scatter in $Y_{2500}$ of $(21pm7)%$ at fixed $Y_{rm X}$, both of which are consistent with previous analyses. We also measure the scaling between $Y_{2500}$ and $M_{2500}$, finding a power-law index of $1.06pm0.12$ and a fractional intrinsic scatter in $Y_{2500}$ at fixed mass of $(25pm9)%$. While recent SZE scaling relations using X-ray mass proxies have found power-law indices consistent with the self-similar prediction of 5/3, our measurement stands apart by differing from the self-similar prediction by approximately 5$sigma$. Given the good agreement between the measured $Y_{2500}$--$Y_{rm X}$ scalings, much of this discrepancy appears to be caused by differences in the calibration of the X-ray mass proxies adopted for each particular analysis.
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