$$CALCET$87\mathrm{.}7\mathrm{.}021.$

$($a$)$ Find the approximations $T_{10},M_{10},$ and $S_{10}$ for ${\displaystyle \underset{0}{\overset{}{}}}33I_{sin}(x)dx.($Round your answers to six decimal places.$)$

$T_{10}=$

$M_{10}=$

$S_{10}=$

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

$E_T=$

$E_M=$

$E_S=$

$($b$)$ Compare the actual errors in part $($a$)$ with the error estimates given by the Theorem about Error Bounds for Trapezoidal and Midpoint Rules and your answers to six decimal places.$)$

$($c$)$ How large do we have to choose $n$ so that the approximations $T_n,M_n,$ and $S_n$ to the integral in part $($a$)$ are accurate to within $0\mathrm{.}00001?$

$n=$ for $T_n$

$n=$ for $M_n$

$n=$ for $S_n$