Integration over a region determined by a given function

How can we compute an integral of the form:

where without writing it as a integral on . I want it so because when A is a very tiny set then the integral function in Matlab applied for the inteval gives a big error.

Here $\phi:(0,1)\to\mathbb{R}$ is a given function. So is (a small number)

[Torsten]

The function to be integrated over (0;1) is

g(x) = f(x) * Indicatorfunction (|phi(x)| < a)

Since this function usually is highly discontinuous, numerical methods (like the ones used in Matlab's "integral") are difficult to apply.

You could try to calculate I_n = trapz(X_n,Y_n) with X_n= linspace(0,1,n) and Y_n = g(X_n) for n = 2,4,8,16,32,... and see whether I_n converges. But note that if phi is very bad behaved, I_n might converge, but not to the correct value of the integral or I_n might not converge although the integral exists.