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

Double integrals of a function f(x, y) over a rectangular region R (continued)

 Objectives:

1. Find a notation for the double integral of a function f(x,y) over a rectangular region R

2. Evaluate a double integral over a rectangular region by writing it as an iterated integral.

Notation.

Remember that the double integral of a function f(x,y) over a rectangular region R is given by:

Iterated Integrals.

The process of calculating double integrals can be lengthy especially if m and n become larger numbers. Therefore it's better to find a way to calculate the integrals without using limits and double sums. The method used consists in breaking the double integral in simple integrals where one integral is evaluated to one variable and the other integral to the other variable, This process is called iterated integral.

The fact that double integrals can be expressed as iterated integral is expressed in Fubini's theorem.

Integrating first with respect to y and then to x to find the area A(x) and then the volume V.

Integrating first with respect to x and then to y to find the area A(y) and then the volume V.

Example

Solution

Since dx = dy, the Fubini's theorem allows to use [a b}or [c d] as upper limit of integration for a function f(x,y) defined over a rectangular region R = [a b] . [c d]. To calculate the double integral, we transform it in iterated integrals.