1. **State the problem:** We have a large cone with height $42$ mm and radius $6$ mm. It is split horizontally into a smaller similar cone at the top with height $14$ mm and radius $2$ mm, and a frustum below it with height $28$ mm. We need to find the volume of the frustum. 2. **Formula for volume of a cone:** $$V = \frac{1}{3} \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Calculate volume of the large cone:** $$V_{large} = \frac{1}{3} \pi (6)^2 (42) = \frac{1}{3} \pi \times 36 \times 42 = 504 \pi$$ 4. **Calculate volume of the smaller cone:** $$V_{small} = \frac{1}{3} \pi (2)^2 (14) = \frac{1}{3} \pi \times 4 \times 14 = \frac{56}{3} \pi$$ 5. **Calculate volume of the frustum:** The frustum volume is the difference between the large cone and the smaller cone volumes: $$V_{frustum} = V_{large} - V_{small} = 504 \pi - \frac{56}{3} \pi = \left(504 - \frac{56}{3}\right) \pi = \frac{1512 - 56}{3} \pi = \frac{1456}{3} \pi$$ 6. **Evaluate the numerical value:** Using $\pi \approx 3.1416$, $$V_{frustum} \approx \frac{1456}{3} \times 3.1416 = 485.3333 \times 3.1416 \approx 1524.3$$ 7. **Final answer:** The volume of the frustum to the nearest integer is **1524** cubic millimeters.