일반 영역의 이중적분

Consider $X\,\text{and}\,Y$ two random variables of probability densities ${p}_{1}(x)$ and ${p}_{2}(y),$ respectively. The random variables $X\,\text{and}\,Y$ are said to be independent if their joint density function is given by $p(x,y)={p}_{1}(x){p}_{2}(y).$ At a drive-thru restaurant, customers spend, on average, $3$ minutes placing their orders and an additional $5$ minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events $X$ and $Y.$ If the waiting times are modeled by the exponential probability densities

$\begin{array}{ccccccc}{p}_{1}(x)=\left\{\begin{array}{ll}\frac{1}{3}{e}^{-x/3} & x\ge 0, \\ 0 & \text{otherwise,}\end{array}\right. & & & \text{and} & & & {p}_{2}(y)=\left\{\begin{array}{ll}\frac{1}{5}{e}^{-y/5} & y\ge 0, \\ 0 & \text{otherwise,}\end{array}\right.\end{array}$

respectively, the probability that a customer will spend less than 6 minutes in the drive-thru line is given by $P[X+Y\le 6]=\iint_{D}p(x,y)dx\,dy,$ where $D=\{(x,y)\}|x\ge 0,y\ge 0,x+y\le 6\}.$ Find $P[X+Y\le 6]$ and interpret the result.