jmc

We want to count the integers from the first one greater than $10.2^3$ to the last one less than $10.3^3$. We note that $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$ for any $a,b$. Using this expansion for $(10+.2{)}^3$ and $(10+.3{)}^3$, we have: $$\begin{array}{rl}(10+.2{)}^3& =10^3+300\cdot .2+30\cdot .04+.008\\ (10+.3{)}^3& =10^3+300\cdot .3+30\cdot .09+.027\end{array}$$We add up, and find the first to be $1061.208$ and the second to be $1092.727$. Thus, we want to count the integers between 1062 and 1092, inclusive; there are $\boxed{31}$ of these.