Calculating Water Volume for a Curved Backyard Pool

You are helping your neighbor determine how much water is needed to fill their new above-ground pool. The pool has an unusual kidney-bean shape that curves smoothly, and the pool floor slopes from a shallow end to a deeper section. The pool manufacturer provided dimensions using one set of variables, but your neighbor’s water bill calculator and the garden hose flow rate chart use standard rectangular measurements.

To figure out the total water volume and how long filling will take, you need to transform the pool’s curved region into a simpler coordinate system, calculate the Jacobian to account for how areas and volumes scale under this transformation, and then evaluate the appropriate integrals. The pool’s curved boundary in the [latex]xy[/latex]-plane is described by the transformation [latex]x = 2u + v[/latex] and [latex]y = u - v[/latex], where the region [latex]G[/latex] in the [latex]uv[/latex]-plane is the rectangle [latex]0 \leq u \leq 3[/latex] and [latex]0 \leq v \leq 2[/latex]. The pool depth at any point is given by [latex]f(x, y) = 1 + 0.1x[/latex] feet.