Properties of double integrals

 Goal: Recognize and use some of the properties of double integrals

The properties of double integrals are very helpful to compute them. Here I list their six properties, Properties 1 and 2 are referred as the linearity of the integral. Property 3 is the additivity of the integral. Property 4 is the monotonicity of the integral, and property 5 is used to find the bounds of the integral. Property6 is used if f(x,y) is a product of two functions g(x) and h(y).  

Example

Solution

Let's apply properties I and II: 

Let's convert the double integrals to iterate:

Integrate with respect to x holding y;

Integrate with respect to y:

Practice

Calculate the following double integral 

over the rectangular region R given by 0 ≤ x ≤ 3., 0≤ y≤2