$2.(5$ pts$)$

Consider the definite integral $\backslash (\backslash$int$\_\{0\}^\{\backslash$pi$\}$ e$^\{\backslash$sin $($x$)\}$ d x $\backslash ).$

$($a$)$ Use the Trapezoid Rule with $\backslash ($ n$=4\backslash )$ to estimate the value of this definite integral.

Show all your work in an organized fashion. You may use your calculator, but no online tools such as WolframAlpha's Trapezoidal Rule Calculator or similar. Report your final answer to $3$ decimal digits.

$($b$)$ Starting with $\backslash ($ n$=4\backslash ),$ use the Method of Successive Approximations, doubling $\backslash ($ n $\backslash )$ in each iteration, to obtain an estimate of the value of this definite integral that is accurate to $3$ decimal digits $($that is$,$ accurate within $\backslash (0\mathrm{.}0001)\backslash ).$

This will take several iterations. It is strongly recommended that you use an online tool such as WolframAlpha's Trapezoidal Rule Calculator. Report your intermediate approximations to $6$ decimal digits, and your final result to $3$ decimal digits.