I was working a problem, and came to a point where it would help greatly if there was a relationship between the following two expressions:
1) The numeric value of $\int_a^b f(x)\,dx$, and
2) The numeric value of $\int_a^bxf(x)\,dx$.
That is, if I know the numeric value for the first integral, but nothing about $f(x)$, is it possible to determine the numeric value for the second?
I've tried experimenting with integration by parts, but that always seems to need the indefinite integral of $F(x)$.
EDIT: In response to Calvin Lin's comment: I'm looking to compute the second integral, so equalities are the type of relationship that I'm looking for.
EDIT 2: In response to JoeHobbit's comment: The particular problem that I'm trying to solve is really this one here. However, this sprung off some other thoughts, not necessarily tied to specific problems.