Multi-variable Integration

Definition with the riemman sums

lim_{n, m \to \infty,} \sum_{j = 1}^{n} \sum_{k = 1}^{m} f (x_{j} y_{k}) \nabla x_{j} \nabla y_{k} = \int

Fubini's Theorem
If $f$ is continuous on the rectangle $R = {a \leq x \leq b, c \leq y \leq d}$ then

\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

Bounded by regions

If no constant bound is given, they can be found by looking at the intersections of the binding functions

The order of the integrals in interchangeable, but some combination of functions are much easier to be integrated when doing a different order of integrals

Bounded by x

Regions bounded by functions of x (type 1 region)

\begin{aligned} R & = {a \leq x \leq b, g_{1} (x) \leq y \leq g_{2} (x)} \\ Area & = \int_{a}^{b} \int_{g_{1} (x_{)}}^{g_{2} (x)} f (x, y) d y d x \end{aligned}

Bounded by y

Regions bounded by functions of y (type 2 region)

\begin{aligned} R & = {c \leq y \leq d, h_{1} (y) \leq x \leq h_{2} (y)} \\ Area & = \int_{c}^{d} \int_{h_{1} (y_{)}}^{h_{2} (y)} f (x, y) d x d y \end{aligned}

Properties of Double Integrals

$\int \int_{R} f (x, y) \pm g (x, y) d A = \int \int_{R} f (x, y) d A \pm \int \int_{R} g (x, y) d A$
$\int \int_{R} c f (x, y) d A = c \int \int_{R} f (x, y) d A$
If $f (x, y) \geq g (x, y), \forall (x, y) \in R^{2}$ then

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

$\int \int_{R} 1 d A$
If $m \leq f (x, y) \leq M, \forall (x, y) \in R^{2}$ then

m A (R) \leq \int \int_{R} f (x, y) d A \leq M A (R)

If $R = R_{1} \cup R_{2}, R_{1} \cap R_{2} = \emptyset$ then

\int \int_{R} f (x, y) d A d A = \int \int_{R_{1}} f (x, y) d A + \int \int_{R_{2}} f (x, y) d A

Multi-variable Change of Variables