It may work if you connect it directly to pin 2 of U1, but I would suggest using the connection in the schematic. C1 provides DC blocking, this is sometimes important with piezo elements (your transducer is a piezo element) because of strain related to temperature change and mounting stress. The 10k "R2" (which should be labeled R1) sets the gain of the amplifier with the other R2 (the 1M pot). Without that, you will always have infinite gain because the ratio will be:
[LATEX] \frac{R2}{R1} &= \frac{R2}{0} &= \infty [/LATEX]
In practice, it is really only around 100,000, because the transducer has some impedance and no amplifier really has infinite gain. But you do not want gain this high, it will amplify too much noise. With R1 = 10k and R2 = 1M you only have a gain of 100:
[LATEX] \frac{R2}{R1} &= \frac{1000000}{10000} &= 100 [/LATEX]