Let $f(x)=\sqrt[\phantom{2}]{1+x^3}.$

$($a$)$ How large must $n$ be in the approximation of ${\displaystyle \underset{0}{\overset{1}{}}}f(x)dx$ by Simpson's

rule to ensure that the absolute error is less than ${10}^{-3}?$

$($b$)$ Estimate the integral using Simpson's rule approximation with the value

of $n$ obtained in part $($a$).$