Vanderbilt Student -- I BOMBED Calc Test After My Billy Joel Duet

Jamming out with Billy Joel is terrible for your report card ... just ask the Vanderbilt freshman whose impromptu duet with the Grammy-winning singer cost him a passing grade on a calculus test.

In case you haven't seen the video -- Billy was at Vanderbilt in January for a Q&A ... and a student named Michael Pollack mustered the testicular fortitude to ask to play piano for Billy while he sang "New York State of Mind." Billy agreed ... and the footage is incredible. Pollack KILLS it.

But the adrenaline rush was so big ... Pollack tells TMZ, he couldn't get himself to concentrate on studying for a big calculus test. The next day, Pollack says he failed the test with a 69.

Pollack tells us, "I'm pretty good at calculus, but it was hard to focus when you perform with your idol ... I have no shame. I got a 69 on the test, compliments of Billy."