Chapter 7.7, Problem 24E

Repeat Exercise 23 for the integral ${\displaystyle {\int }_{-1}^1\sqrt{4-x^3}dx}$.

The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for $\left|f^4\left(x\right)\right|$ by hand. But computer algebra systems have no problem computing $f^{\left(4\right)}$ and graphing it, so we can easily find a value for K from a machine graph. This exercise deals with approximations to the integral $I={\displaystyle {\int }_0^{2\pi }f\left(x\right)dx}$, where $f\left(x\right)=e^{\cos x}$.

(a) Use a graph to get a good upper bound for $\left|f^{″}\left(x\right)\right|$.

(b) Use $M_{10}$ to approximate I.

(c) Use part (a) to estimate the error in part (b).

(d) Use the built-in numerical integration capability of your CAS to approximate I.

(e) How does the actual error compare with the error estimate in part (c)?

(f) Use a graph to get a good upper bound for $\left|f^{(4)}\left(x\right)\right|$.

(g) Use $S_{10}$ to approximate I.

(h) Use part (f) to estimate the error in part (g).

(i) How does the actual error compare with the error estimate in part (h)?

(j) How large should n be to guarantee that the size of the error in using $S_n$ is less than 0.0001?