jmc

By Vieta's Formulas, we know that $a$ is the sum of the three roots of the polynomial $x^3-a{x}^2+bx-2010$. Again Vieta's Formulas tells us that $2010$ is the product of the three integer roots. Also, $2010$ factors into $2\cdot 3\cdot 5\cdot 67$. But, since there are only three roots to the polynomial, two of the four prime factors must be multiplied so that we are left with three roots. To minimize $a$, $2$ and $3$ should be multiplied, which means $a$ will be $6+5+67=\boxed{78}.$