$$ \newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\dx}{\text{ dx}}
\newcommand{\rang}{\text{rang}}
\newcommand{\s}{\ \ \ \ \ \ }
\newcommand{\arrows}{\s \Leftrightarrow \s}
\newcommand{\Arrows}{\s \Longleftrightarrow \s}
\newcommand{\arrow}{\s \Rightarrow \s}
\newcommand{\c}{\bcancel}
\newcommand{\v}[2]{
\begin{pmatrix}
#1 \\
#2 \\
\end{pmatrix}
}
\newcommand{\vt}[3]{
\begin{pmatrix}
#1 \\
#2 \\
#3 \\
\end{pmatrix}
}
\newcommand{\stack}[2]{
\substack{
#1 \\
#2
}
}
\newcommand{\atom}[3]{
\substack{
#1 \\
#2
}
\ce{#3}
}
$$
Change of Variables
Used to get of non-constant boundaries of integrals.
We define a function to map another, more convenient, space to the original space:
$$ \int_{g(D)} f(x,y) = \int_D f(g(u,v)) \cdot ||J|| $$
$f(x,y)$: The original function
$g(x,y)$: The function in the more convenient space ($g$ must be injective)
$||J||$: The absolute value of the determinant of the jacobian of $g(u, v)$ at the point $(u,v)$
Other Coordinate Systems
It is often a good idea to translate an integral to another known coordinate system, with a known jacobian determinant. Here are some options: