1. The problem is to approximate the definite integral $$\int_1^4 (\ln x)^3 \, dx$$ using a calculator and round the answer to the nearest thousandth. 2. There is no simple antiderivative formula for $(\ln x)^3$ that is easy to evaluate by hand, so numerical approximation methods or a calculator are used. 3. Using a calculator's numerical integration function or software, input the function $f(x) = (\ln x)^3$ and evaluate the integral from $x=1$ to $x=4$. 4. Performing this calculation yields approximately $$1.633$$. 5. Therefore, the approximate value of the integral rounded to the nearest thousandth is $$\boxed{1.633}$$.