Let $f(x)=\cos(x^2).$

Use (a) the Trapezoidal Rule and (b) the Midpoint Rule to approximate the integral $\int_{0}^{\,1} {f(x)}\, dx$ with $n = 8$. Give each answer correct to six decimal places.

$T_{8} =$

$M_{8} =$

(c) Use the fact that $|f''(x)| \le 6$ on the interval $[0,1]$ to estimate the errors in the approximations from part (a). Give each answer correct to six decimal places.

Error in $T_{8}$ =

Error in $M_{8}$ =

(d) Using the information in part (c) and the error formulas, how large do we have to choose $n$ so that the approximations $T_n$ and $M_n$ are accurate to within 0.00001? (Your answer must be a whole number.)

For $T_n:$ $n =$

For $M_n:$ $n =$