Integration

Theorem: for \(D \subseteq \mathbb{R}^n\) (\(n=2\) for simplicity) with \(D = X \times Y\), let \(f: D \to \mathbb{R}\) then we have

\[ \iint_D f = \int_X \Big(\int_Y f(x,y)dy \Big)dx = \int_Y \Big(\int_X f(x,y)dx \Big)dy \]

implying that order can be interchanged, this is true for \(n \in \mathbb{N}\).

Proof:

Will be added later.

Iteration of integrals

Theorem: for \(D \subseteq \mathbb{R}^n\) (\(n=2\) for simplicity) bounded and piecewise smooth boundary, let \(f: D \to \mathbb{R}\) be bounded and continuous. Let \(R\) be a rectangle with \(D \subseteq R\) then

\[ \iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad &\mathbf{x} \in D, \\ 0 \quad &\mathbf{x} \notin D. \end{cases} \]

Proof:

Will be added later.

Coordinate transformation for integrals

Theorem: for \(D \subseteq \mathbb{R}^n\) (\(n=2\) for simplicity) bounded and piecewise smooth boundary, let \(f: D \to \mathbb{R}\) be bounded and continuous and let \(\phi: D \to \mathbb{R}^n\) be continuously differentiable and injective, define

\[ E := \phi(D), \]

then we have

\[ \iint_D f = \iint_E f \circ \phi \;\Big|\mathrm{det} \big(D_\phi \big) \Big|, \]

with \(D_\phi\) the Jacobian of \(\phi\).

Proof:

Will be added later.

Example

Let \(D = \big\{(x,y) \in \mathbb{R}^2 \;\big|\; x^2 + y^2 \leq 4 \land 0 \leq y \leq x \big\}\) and let \(\phi: D \to \mathbb{R}^2\) be given by

\[ \phi(r,\theta) = \begin{pmatrix} r\cos \theta \\ r\sin \theta \end{pmatrix}, \]

define \(E := \phi(D) = [0,2] \times [0, \frac{\pi}{4}]\). Then \(E\) is a rectangle which can be more easily integrated.