Evaluate the integral. $($Remember the constant of integration.$)$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}3x\sqrt[\phantom{2}]{1-x^4}dx$

$x$ Check your answer by taking the derivative.

Do the following with the given information.

${\displaystyle \underset{0}{\overset{1}{}}}39cos(x^2)dx$

$($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.$)$

$|E_T|$

$|E_M|$

Nice work!

$($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.$)$

$n$

for $T_n$

$n$

for $M_n$