PX153 - I6 - volume integrals

so far, cartesian coordinates have been used. a more general 2D integral:

A = \iint_{R} f (\vec{r}) . d A

where, $\vec{r} = (x, y, z)$

for volume integrals:

I = ∭_{V} f (\vec{r}) . d V

where, $d V = d x . d y . d z$ in cartesian coordinates

eg: what is the mass of a cube, $- L \leq x, y, z \leq L$ , with $ρ (\vec{r}) = ρ_{0} (1 + x)$

\begin{aligned} M & = \int_{- L}^{L} \int_{- L}^{L} \int_{- L}^{L} ρ_{0} (1 + x) . d x . d y . d z \\ = \int_{- L}^{L} \int_{- L}^{L} 2 L ρ_{0} . d y . d z \\ = 2 L ρ_{0} [y]_{- L}^{L} [z]_{- L}^{L} \\ = ρ_{0} (2 L)^{3} \\ ∴ M & = ρ_{0} V \end{aligned}