jmc

First, the $2002$th positive even integer is $4004$, so we are looking for the number of digits used when the positive even integers less than or equal to $4004$ are written. Split this problem into cases. It's easy to see that there are four positive even integers with one digit: $2$, $4$, $6$, and $8$. Beginning with $10$ and ending with $98$, there are $\frac{98-10}{2}+1=45$ positive even numbers with two digits. Beginning with $100$ and ending with $998$, there are $\frac{998-100}{2}+1=450$ positive even numbers with three digits. Finally, beginning with $1000$ and ending with $4004$, there are $\frac{4004-1000}{2}+1=1503$ positive even numbers with four digits. So, our answer is $4+2\cdot 45+3\cdot 450+4\cdot 1503$, which equals $4+90+1350+6012$. Thus, there are $\boxed{7456}$ digits used.