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

(a) Compute $\int_{-1}^{\,2} {f(x)}\,dx.$

(b) Compute the approximations $T_n,$ $M_n,$ and $S_n$ for $n$ = 6 and 12 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.

$T_6 =$

$|E_T| =$

$M_6 =$

$|E_M| =$

$S_6 =$

$|E_S| =$

$T_{12} =$

$|E_T| =$

$M_{12} =$

$|E_M| =$

$S_{12} =$

$|E_S| =$

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