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The biographical sketch of Rabbi Moshe Feinstein that appears in Igrot Moshe volume 8 describes the following (my translation):

The Communist takeover meant the schools taught against religion. Some of the older Jewish students would argue back, citing teachings of their rabbi [Moshe Feinstein]. One infuriated teacher (herself Jewish) finally replied, "What does that rabbi of yours know, if he can't solve this calculus problem?!"

Eventually the problem was brought to Rabbi Feinstein; who requested a calculus textbook, read it, and then solved it.

Have any details about this story (other than the ones printed there) survived to this day?

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Well, the original you paraphrased does imply it's differential calculus. – msh210 Jun 13 '11 at 19:01
I suggest you ask him: daattorah.blogspot.com – simchashatorah Jun 14 '11 at 3:04
Perhaps this should be asked on Math.SE ;) – AviD Jun 14 '11 at 7:10

There is a very slightly different account here, which places the story in the context of religious persecution and Rav Moshe's ostensible motivations for leaving Russia.

But something tells me by "details about this story" you mean "specific contents of the calculus problem", in which case this is no help. . .

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