Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 11G / ii / Solution

True. An affine change of variables carries $[a,b{]}^2$ to $[0,1{]}^2$. Equivalently, polynomials in $s,t$ are uniformly dense on $[a,b{]}^2$ by the rescaled Bernstein construction. The moment assumptions annihilate every polynomial, so approximating $f$ itself again gives $\int f^2=0$ and hence $f=0$.