Let $f(x)=x^3.$

(a) Compute $\int_{0}^{\,1} {f(x)}\,dx.$

(b) Compute the approximations $L_n,$ $R_n,$ $T_n,$ and $M_n$ for $n$ = 4, 8, and 16 for the integral in part (a). For each of these, compute the corresponding absolute error.

Note: Make sure all answers are correct to six decimal places.

$L_4 =$

$|E_L| =$

$R_4 =$

$|E_R| =$

$T_4 =$

$|E_T| =$

$M_4 =$

$|E_M| =$

$L_8 =$

$|E_L| =$

$R_8 =$

$|E_R| =$

$T_8 =$

$|E_T| =$

$M_8 =$

$|E_M| =$

$L_{16} =$

$|E_L| =$

$R_{16} =$

$|E_R| =$

$T_{16} =$

$|E_T| =$

$M_{16} =$

$|E_M| =$

Note: Observe what happens to the errors when $n$ is doubled.