Double Integrals over General Domains

# Double Integrals over General Domains

We have just looked at Double Integrals over Rectangles. Recall that if $z = f(x, y)$ is a two variable real-valued function and let $R = [a, b] \times [c, d] \subseteq D(f)$ be a rectangular subset of the domain. Then the double integral of $f$ over $R$ (provided this limit exists) is:

(1)
\begin{align} \iint_{R} f(x, y) \: dA = \lim_{m, n \to \infty} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*) \Delta A \end{align}

Sometimes we may be more interested in computing double integrals over subsets of the domain that aren't necessarily rectangles and instead some other shape. Let $D \subseteq D(f)$ be a subset of the domain that is not necessarily a rectangle, and inscribe $D$ within a rectangle $R$ whose side lengths are parallel to the $x$ and $y$ axes.

We will define the two variable real-valued function $\hat{f} (x, y)$ as follows:

(2)
\begin{align} \hat{f} (x, y) = \left\{\begin{matrix} f(x,y) & \mathrm{if} \: (x,y) \in D \\ 0 & \mathrm{if} \: (x,y) \not \in D \end{matrix}\right. \end{align}

We are now ready to define the double integral of $f$ over $D$.

 Definition: Let $z = f(x, y)$ be a two variable real-valued function that is bounded on $D \subseteq D(f)$ and let $R$ be a rectangle whose sides are parallel to the $x$ and $y$ axes such that $D \subseteq R$. Let $f$ be integrable over $R$, and let $\hat{f} = \left\{\begin{matrix} f(x,y) & \mathrm{if} \: (x,y) \in D \\ 0 & \mathrm{if} \: (x,y) \not \in D \end{matrix}\right.$. Then the Double Integral of $f$ over $D$ is defined to be $\iint_D f(x,y) \: dA = \iint_{R} \hat{f} (x,y) \: dA$.

Note that $\iint_D f(x,y) \: dA = \iint_{R} \hat{f} (x,y) \: dA$ since for all $(x, y) \not \in D$, $\hat{f} (x, y) = 0$ which does not contribute any value to the double integral.