We observed a 10x20 pc region of the molecular cloud M17 in the 12CO and 13CO J=3-2 and J=2-1 transitions to determine their global behavior and to assess the reliability of using ratios of CO line intensities integrated over an entire cloud to determine the physical conditions within the cloud. Both the 12CO/13CO J=2-1 and J=3-2 line ratios correlate with the 13CO integrated intensity, with smaller line ratios observed at locations with large integrated intensities. This correlation is likely due to variations in the column density from one position to another within M17. The 12CO and 13CO (J=3-2/J=2-1) line ratios show no significant variation from place to place within M17, even on the peak of the photon-dominated region. A Large Velocity Gradient analysis of globally averaged line ratios gives results in reasonable agreement with the results obtained for individual lines-of-sight through the cloud, which suggests that the typical physical conditions in a molecular cloud can be determined using CO line ratios integrated over the entire cloud. There appears to be a clear trend of increasing 12CO/13CO J=2-1 and J=3-2 line ratios as one moves from Galactic molecular cloud cores to entire Galactic molecular clouds to normal galaxies. The most likely explanation of the high line ratios for normal galaxies is a significant contribution to the CO emission by low column density material, such as diffuse molecular clouds or the outer envelopes of giant molecular clouds.
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