A Game of Brackets!

$\large \frac{\displaystyle \int_{0}^{91} \lfloor \lfloor x^{2} \rfloor^{\frac{1}{2}}\rfloor dx }{\displaystyle \int_{0}^{20} \lfloor \lfloor x^{2} \rfloor^{\frac{1}{2}}\rfloor dx }$

Let this ratio be one that condenses to $\displaystyle \large \frac{9}{2} \frac{\alpha}{\beta}$ where $\large \alpha$ and $\large \beta$ are coprime positive integers. Find the product $\alpha \beta$ .