Double Integral Over a Rectangle

Basic Definition of a Double Integral

Assume $R \subset R^{2}$ is a closed rectangle where $R = [a, b] \times [c, d]$
(open would be something like $(a, b] \times [c, d]$ )

Take small segments along the two axes so that

x_{j + 1} - x_{j} = \frac{b - a}{n}, y_{k + 1} - y_{k} = \frac{d - c}{n}

A continuous function on a closed rectangle is always bounded where $\exists M > 0$ such that $- M < f (x) < M$

Let $R_{j k}$ be the rectangular partition $[x_{j}, x_{j + 1}] \times [y_{k}, y_{k + 1}]$

The double integral can be defined at the sum

S_{n} = \sum_{j, k = 0}^{n - 1} f (c_{j k}) Δ x Δ y = \sum_{j, k = 0}^{n - 1} f (c_{j k}) Δ A

Where $c_{j k}$ is an arbitrary point in $R_{j k}$

where we take $lim_{n \to \infty}$ , if the limit converges to the same value no matter the choice of $c_{j k}$ (for example the left-most point versus the right-most point) then $f$ is integrable over $R$

and we write this limit as

\int \int_{R} f (x, y) d A = \int \int_{R} f (x, y) d x d y = \int \int_{R} f d x d y

Properties

Any continuous function defined on a closed rectangle is integrable
A real-valued function is also integrable if it is discontinuous only a finite region but still bounded

Linearity

\int \int_{R} [f (x, y) + g (x, y)] d A = \int \int_{R} f (x, y) d A + \int \int_{R}, g (x, y) d A .

Homogeneity

\int \int_{R} c f (x, y) d A = c \int \int_{R} f (x, y) d A

Monotonicity

f (x, y) \geq g (x, y) ⟹ \int \int_{R} f (x, y) d A \geq \int \int_{R} g (x, y) d A

Additivity

if $R_{i} i = 1, \dots, m$ are partitioned rectangles such that $f$ is bounded and integrable over each $R_{i}$ and their union is also a rectangle $R$ than:

\int \int_{R} f (x, y) d A = \sum_{i = 1}^{m} \int \int_{R_{i}} f (x, y) d A

Fubini's Theorem

Let $f$ be bounded and continuous in each partition of $R = [a, b] \times [c, d]$ :

\int \int_{R} f (x, y) d A = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x = \int_{c}^{d} \int_{a}^{b} f (x, y) d x d y

Basic idea of the proof

\begin{aligned} \int \int_{R} f (x, y) d A = lim_{n \to \infty} \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - 1} f (c_{j k}) (y_{k + 1} - y_{k}) (x_{j + 1} - x_{j}) \\ by MVT \forall c_{j k} = (x, y) \in R_{i} \exists y^{*} \frac{1}{y_{k + 1} - y_{k}} \int_{y_{k}}^{y_{k + 1}} f (x, y) d y = f (x, y *) \\ so \int_{y_{k}}^{y_{k + 1}} f (c_{j k}) d y = f (x, y *) (y_{k + 1} - y_{k}) \\ \int \int_{R} f (x, y) d A = lim_{n \to \infty} \sum_{j = 0}^{n - 1} f (x, y *) (y_{k + 1} - y_{k}) (x_{j + 1} - x_{j}) = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x \end{aligned}