Decomposing regions into smaller regions in the calculation of double integrals

 Decomposition of regions into smaller regions in the calculation of double integrals

Goal: Decompose a region into smaller regions in the calculation of double integrals.

The properties of double integrals over rectangular regions are valid for the the calculation of double integrals over non rectangular bounded region on a plane. The property 3 of double integrals over rectangular region states:

The following theorem establishes the same property for double integrals over non rectangular regions on a plane.

Theorem 

Suppose the region D can be expressed as D = D₁∪D₂ where D₁ and D₂ do not overlap except at their boundaries.

Example

Solution

Here D₁ is type 1 and D₂ and D₃ are type 2. Hence: