Question

A special liquid is held in a tank described as (x 2 + y 2 ) ≤ z ≤ 1 in a Cartesian coordinate system. Assume that the density of the liquid at point (x, y, z) is rho(x, y, z) = (2 − z 2 ), provided that the point is actually in the tank. Find the total weight of the liquid by integrating rho dx dy dz.

Answer 1

Since
`\rho=\frac mV` (density = mass/volume), we can get the mass/weight of the liquid by integrating the density
`\rho(x,y,z)` over the interior of the tank. This is done with the integral

`\displaystyle\int_(-1)^1\int_(-√(1-x^2))^(√(1-x^2))\int_(x^2+y^2)^1(2-z^2)\,\mathrm dz\,\mathrm dy\,\mathrm dx`

which is more readily computed in cylindrical coordinates as

`\displaystyle\int_0^(2\pi)\int_0^1\int_(r^2)^1(2-z^2)r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{\frac{3\pi}4}`