$(20$ marks total$)($Monte Carlo integration$).$ Given the integral

${\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{-1}{\overset{1}{}}}{x}^{2}ydydx$

$($a$).(10$ marks$)$ Write a well$-$documented MATLAB script where you use the Monte Carlo

simulations to estimate the integral

$($b$.(10$ marks$)$ Write a well$-$documented MATLAB script where you apply the antithetic

variates method. Run both MATLAB codes with the number of simulations, $N=$

$100,500,2000,3000,$ compare the results and make your observations.