mass of a lamina

The total mass of a flat plate with variable density, $m=\iint_R\rho(x,y)\,dA$ .

The mass of a lamina (a flat plate) occupying region $R$ with density $\rho(x,y)$ is $m=\iint_R\rho(x,y)\,dA$ — read the same integrand that was previously a height as a density instead, and the double integral answers a different question with identical machinery. The constant-density special case $\rho(x,y)=k$ reduces $m=k\cdot A(R)$ to the familiar "mass equals density times area," recovered as one instance of the general formula.

Examples

A square plate $[0,1]\times[0,1]$ with density $\rho(x,y)=x+y$ has mass $m=\int_0^1\!\int_0^1(x+y)\,dy\,dx=1$ kg.