Given.

${\displaystyle \underset{0}{\overset{1}{}}}7cos(x^2)dx$

Do the following.

$($a$)$ Find the approximations $T_8$ and $M_8$ for the given integral. $($Round your answer to six decimal places.$)$

$T_8=$

$M_8=$

$($b$)$ Estimate the errors in the approximations $T_8$ and $M_8$ in part $($a$).($Use the fact that the range of the sine and cosine functions is bounded by $-1$ to estimate the maximum error. Round your answer to seven decimal places.$)$

$|E_T|$

$|E_M|$

$($c$)$ How large do we have to choose $n$ so that the approximations $T_n$ and $M_n$ to the integral are accurate to within $0\mathrm{.}0001?($Use the fact that the range of the sine and cosine functions is bounded by $-1$ to estimate the maximum error.$)$

$n$

$n$