Double Integration

Basic Integration

Recall that the definite integral is defined as the limit of the Riemann sum:

$${\int }_a^{b}f(x)dx=\underset{||\Delta x_i||\to 0}{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x_i$$

where $a=x_1<x_2<\cdots <x_n=b$, $\Delta x_i=x_{i+1}-x_i$ and $x_i^{\ast }\in [x_i,x_{i+1}]$.

Definition of Double Integral

Suppose we have a function $f(x,y)$. How do we define the volume under the curve for any given closed region $R$?

The double integral of $f(x,y)$ over region $R$ is defined as:

$${\iint }_{R}f(x,y)dA=\underset{||\Delta A_i||\to 0}{\lim }\sum _{i=1}^{n}f(x_i^{\ast },y_i^{\ast })\Delta A_i$$

where $\{R_i|i\in I\}$ is a partition of $R$, $\Delta A_i$ is the area of the sub-region $R_i$ and $(x_i^{\ast },y_i^{\ast })\in R_i$.

Properties

1. If ${\iint }_{R}f(x,y)dA$ exists and $c\in \mathbb{R}$ then:

$${\iint }_{R}cf(x,y)dA=c{\iint }_{R}f(x,y)dA$$

2. If ${\iint }_{R}f(x,y)dA$ and ${\iint }_{R}g(x,y)dA$ then

$${\iint }_{R}f(x,y)+g(x,y)dA={\iint }_{R}f(x,y)dA+{\iint }_{R}g(x,y)dA$$

3. If $R$ is partitioned by $R_1$ and $R_2$ that have piecewise-smooth boundaries and ${\iint }_{R}f(x,y)dA$ exists then:

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

1. If the area of $R$ is defined then it is given by:

$$\text{area of }R={\iint }_{R}dA$$