$($Maple Question$).$ In this question we will explore the definition of a definite integral. Consider the function

$f(x)=\frac{1}{\sqrt[\phantom{2}]{2}}{e}^{-\frac{x^2}{2}},on$ the interval $[-5$;$5].$

The area under the graph is defined as ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=\underset{n}{lim}(f(x_1)+f(x_2)+$dots$+f(x_n))x$

a$)$ Open a new Maple worksheet and on the first line type: $>$ restart:with$($plots$)$;

b$)$ Use Maple to plot $f(x)$ on the interval $-5,5$ :

$)=(-5.\mathrm{.}5$

c$)$ Compute the exact area: $)=(-5.\mathrm{.}5$;

d$)$ Now we will approximate the area with rectangles. Plot the boxes and compute the area under the graph using $n=10$ rectangles with

$)=(-5.\mathrm{.}5,10))=((-5.\mathrm{.}5,10$

e$)$ Repeat the same procedure as d$)$ with $n=50,n=100$ and $n=250$ boxes and fill out the following table:

$\backslash$table$[[n,10,50,100,250],[$Area:$,,,,]]$

f$)$ Compare the calculated area $S$ with your approximations and explain the results. Submit your printout for b$),$ c$),$ and write your answer for e$)$ and f$)$ by hand on the printout.