$(a)$ Find the approximations $T_{10},M_{10},$ and $S_{10}$ for ${\displaystyle \underset{0}{\overset{}{}}}32sin(x)dx.(Round$ your answers $to$ six decimal places.$)$

$T_{10}=63\mathrm{.}472753$

$M_{10}=$

$S_{10}=64\mathrm{.}003505$

Find the corresponding errors $E_T,E_M,$ and $E_S.(Round$ your answers $to$ six decimal places.$)$

$E_T=0\mathrm{.}527247$

$E_M=0\mathrm{.}330734$

$E_S=$

$$

bound for Simpson's rule.$(Round$ your answers $to$ six decimal places.$)$

$|E_T|0\mathrm{.}826833$

$|E_M|$

$|E_S|$

$(c)$ Using the values $ofK$ from part $(b),$ how large $dowe$ have $to$ choose $nso$ that the approximations $T_{n^{\text{'}}}{M}_{n^{\text{'}}}$ and $S_{n}to$ the integral $in$ part $(a)$ are accurate $to$ within $0\mathrm{.}00001?$

For $T_{n^{\text{'}}}n=,.$

For $M_{n^{\text{'}}}n=,x.$

For $S_{n^{\text{'}}}n=,x.$