Integral Region Additivity Property

Region

For Region $R={\cup }_{i=1}^n{R}_i$ Then:

$$\int {\int }_{{}_R}f(x,y)dA=\sum _{i=1}^n\int {\int }_{{}_{R_i}}f(x,y)dA$$

2D Region Specific

For a 2D Region $R=R_1\times R_2$

$$\int {\int }_{R}f(x,y)dA=\int {\int }_{R_1}f(x,y)dA+\int {\int }_{R_2}f(x,y)dA$$