Question

Find the mass of the rectangular prism 0<=x<=3, 0<=y<=1, 0<=z<=2, with density function ρ(x,y,z)=x

Answer 1

The mass is given by:
\[M=\int\limits_V \rho(x,y,z)dV\]
So:
\[M=\int\limits_0^2 \int\limits_0^1 \int\limits_0^3x dx dy dz=\frac{1}{2}\int\limits_0^2 \int\limits_0^1 x^2|_0^3dy dz=4 \int\limits_0^2dz \int\limits_0^1dy=4*2*1=8\]

Answer 2

I'm sorry, that should be:
\[\frac{1}{2}\int\limits_0^2dz \int\limits_0^1dy x^2|_0^3=\frac{9}{2}*2*1=9\]