Do the following with the given information.

${\displaystyle \underset{0}{\overset{1}{}}}28cos(x^2)dx$

$($a$)$ Find the approximations $T_8$ and $M_8$ for the given integral.

$($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.$)$

$($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.$)$

Step $1$

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

We will first find the approximation $T_g$ for the following integral.

${\displaystyle \underset{0}{\overset{1}{}}}28cos(x^2)dx$

Recall that the Trapezoidal Rule is given by the following, where $x=\frac{b-a}{n}$ and $x_i=aix.$

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=T_n=\frac{x}{2}[f(x_0)2f(x_1)$ cdots $2f(x_{n-1})f(x_n)]$

To apply the Trapezoidal Rule to find $T_8$ for the given integral, we use the following values.

$a=0,0$

$b=1,1$

$n=8,8$

$x=0\mathrm{.}125>\frac{1}{8}$

Step $2$

We have determined that $a=0$ and $x=\frac{1}{8},$ which allows us to determine all the needed values of $x_i=aix.$

$x_0=00(\frac{1}{8})=0$

$x_1=01(\frac{1}{8})=\frac{1}{8}$

$x_2=0\mathrm{.}25$

$x_3=0\mathrm{.}375$

$x_4=0\mathrm{.}5$

$x_5=0\mathrm{.}625$

$x_6=0\mathrm{.}75$

$x_7=0\mathrm{.}875$

$x_8=08(\frac{1}{8})=\frac{1}{2}$

Step $3$

Using the endpoints of the subintervals that we found allows us to write $T_8$ as follows, where $f(x)=28cos(x^2).$

$T_n=\frac{x}{2}[f(x_0)2f(x_1)$ cdots $2f(x_{n-1})f(x_n)]$

$T_8=\frac{(\frac{1}{8})}{2}[f(0)2f(\frac{1}{8})2f(\frac{1}{4})2f(\frac{3}{8})2f(\frac{1}{2})2f(\frac{5}{8})2f(\frac{3}{4})2f(\frac{7}{8})f(1)]$

Evaluating the above and rounding the result to six decimal places gives the following result.

$T_8=$