1. The problem states that region $R$ is bounded by the curve $y=f(x)$ and the $x$-axis between $x=a$ and $x=b$. 2. The solid $S$ has cross-sections perpendicular to the $x$-axis that are squares with side length equal to $f(x)$. 3. The volume $V$ of the solid is found by integrating the area of each square cross-section along the $x$-axis from $a$ to $b$. 4. The area of each square cross-section is $[f(x)]^2$ because the side length is $f(x)$. 5. Therefore, the volume is given by the integral $$V=\int_a^b [f(x)]^2 \, dx.$$ 6. This matches exactly the formula given in the problem statement. 7. Hence, the statement is \textbf{True}.