$($a$)$ Find the approximations $T_{10},M_{10},$ and $S_{10}$ for ${\displaystyle \underset{0}{\overset{}{}}}29sin(x)dx.($Round your answers to six decimal places.$)$

$T_{10}=57\mathrm{.}522183$ Nice job.

$M_{10}=$

$S_{10}=$ Nice job.

Fantastic work!

Find the corresponding errors $E_T,E_{M^{\text{'}}}$ and $E_{S^{\text{'}}}($Round your answers to six decimal places.$)$

$E_T,=$ Fantastic!

$E_M,=$ That's right!

$E_S,=$ Nicely done! error bound for Simpson's rule.$($Round your answers to six decimal places.$)$

$|E_T|$

$|E_M|$ Awesome job!

$|E_S|$ Excellent job!

$($c$)$ Using the values of $K$ from part $($b$),$ how large do we have to choole $n$ so that the approximations $T_n,M_n,$ and $S_n$ to the integral in part $($a$)$ are accurate to within $0\mathrm{.}00001?$

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

For $M_{n^{\text{'}}}n=$

For $S_{n^{\text{'}}}n=$