5.1 Double integrals

Consider a rectangular box $B$ in $0xyz$ space, with base $R$ given by $a\leq x\leq b$ , $c\leq y\leq d$ and height above the $0xy$ plane represented by $z$. We partially fill the box with sand, where the height of the sand above the point $(x,y)$ is represented by a continuous function $z=f(x,y)$. The volume of the sand is the double integral

which we interpret as follows. We split up the rectangle into $N^{2}$ smaller rectangles $R_{ij}$ of sides $h$ by $k$ by introducing a grid with vertices $(x_{i},y_{j})=(a+hi,c+kj)$ for $1\leq i,j\leq N$ with $b=a+Nh$ and $d=c+Nk$.