Let $f(x)=\sqrt{4-x^3}$ and $I=\int_{-1}^{\,1} {f(x)}\,dx.$

(a) Use $M_{10}$ to approximate I to six decimal places.

$M_{10} =$

(b) Use the fact that $|f''(x)| \le \frac{5}{4}\sqrt{3}$ on $[-1,1]$ to obtain an upper bound on the absolute error $|E_M|$ of the approximation from (a). Make sure your answer is correct to six decimal places.

$|E_M| \le$

(c) Use $S_{10}$ to approximate I to six decimal places.

$S_{10} =$

(d) Use the fact that $|f^{(4)}(x)| \le \frac{167}{16}\sqrt{3}$ on $[-1,1]$ to obtain an upper bound on the absolute error $|E_S|$ of the approximation from (c). Make sure your answer is correct to six decimal places.

$|E_S| \le$

(e) Use a CAS to approximate I to six decimal places.

$I \approx$

(f) Which of the two approximations $M_{10}$ and $S_{10}$ is better?

A. Both are equally good approximations.
B. $M_{10}$
C. $S_{10}$

(g) Using the information given in part (d) and the appropriate error formula in your textbook, how large do we have to choose $n$ to ensure that the approximation $S_n$ is accurate to within 0.0001? (Your answer must be a whole number.)

$n =$