The terminal gates are not actually located continuously from $0$ to

$1,$ as we assumed in part $($a$).$ There are only a finite number of gates and they

are likely to be equally spaced. Suppose there are n$+1$ gates located $1/$n units

apart from one end of the terminal $($xo $=0)$ to the other $($xn $=1).$ Assuming

that all pairs $($i$,$j$)$ of arrival and departure gates are equally likely, show that

Average distance between gates $=1$

$($n $+1)2$

n$$

i$=0$

n$$

j$=0$

$|$ i

n $$ j

n $|.$

Identify this sum as approximately $($but not exactly$)$ a Riemann sum with n

subdivisions for the integrand used in part $($a$).$ Compute this sum for n $=$ a and

n $=$ a $+$ b and compare to the answer of part $($a$)$

Note: Use the values of the constants that have been assigned to you for

a$,$ b$,$ c$,$ d$,$ and e