Volumes?

$\begin{aligned} V & = \int_{0}^{1} \int_{x^3}^{x^2} \int_{0}^{t^2} \ dy \ dt \ dx\\ \\ & = \int_{0}^{1} \int_{x^3}^{x^2} \left[ y \right]_{0}^{t^2} \ dt \ dx\\ \\ & = \int_{0}^{1} \int_{x^3}^{x^2} t^2 \ dt \ dx\\ \\ & = \int_{0}^{1} \left[\dfrac{1}{3} t^3\right]_{x^3}^{x^2} \ dx\\ \\ & = \dfrac{1}{3} \int_{0}^{1} \left(x^6 - x^9\right) \ dx\\ \\ & = \dfrac{1}{3} \left[\dfrac{1}{7} x^7 - \dfrac{1}{10} x^{10}\right]_{0}^{1}\\ \\ & = \dfrac{1}{3} × \dfrac{\left(10 - 7\right)}{70}\\ \\ & = \dfrac{1}{70}\\ \implies A & = 0.01428\\ \implies \left \lfloor 1000A \right \rfloor & = \color{#69047E}{\boxed{14}} \end{aligned}$