The shape of a swimming pool is described by the function y=-x^2 + 9. The area between the boundaries x=-3 and x=3 is 36 square meters. How can I calculate the volume if the bath is 1 m deep on the widest side (the straight one) and 1.8 m on the round side, where the depth decreases evenly?

## Answer

This can be done with a double integral over the base (the part inside the parabola y = 9 – x^{2}) where the depth is used as a function f(x,y). This way you determine the volume below f(x,y) and above the ground plane. So we take the depth as positive.

That depth increases linearly from 1m for y=0 to 1.8m for y = 9, and can therefore be described as: depth(x,y) = 1 + 0.8*y/9 so it only depends on y, not on X

the integral is then: see attached image

…. = 47.52 cubic meters

## Answered by

#### prof.dr. Paul Hellings

Department of Mathematics, Fac. IIW, KU Leuven

Old Market 13 3000 Leuven

https://www.kuleuven.be/

.