### Key Terms

- double integral
- of the function $f(x,y)$ over the region $R$ in the $xy$-plane is defined as the limit of a double Riemann sum, $\underset{R}{\iint}f(x,y)dA}=\underset{m,n\to \infty}{\text{lim}}{\displaystyle \sum _{i=1}^{m}{\displaystyle \sum _{j=1}^{n}f({x}_{ij}^{*},{y}_{ij}^{*})\text{\Delta}A.}$

- double Riemann sum
- of the function $f(x,y)$ over a rectangular region $R$ is $\sum _{i=1}^{m}{\displaystyle \sum _{j=1}^{n}f({x}_{ij}^{*},{y}_{ij}^{*})\text{\Delta}A}$ where $R$ is divided into smaller subrectangles ${R}_{ij}$ and $({x}_{ij}^{*},{y}_{ij}^{*})$ is an arbitrary point in ${R}_{ij}$

- Fubini’s theorem
- if $f(x,y)$ is a function of two variables that is continuous over a rectangular region $R=\{(x,y)\in {\mathbb{R}}^{2}|a\le x\le b,c\le y\le d\},$ then the double integral of $f$ over the region equals an iterated integral, $\underset{R}{\iint}f(x,y)dy\phantom{\rule{0.2em}{0ex}}dx}={\displaystyle {\int}_{a}^{b}{\displaystyle {\int}_{c}^{d}f(x,y)dx\phantom{\rule{0.2em}{0ex}}dy}}={\displaystyle {\int}_{c}^{d}{\displaystyle {\int}_{a}^{b}f(x,y)dx\phantom{\rule{0.2em}{0ex}}dy}$

- improper double integral
- a double integral over an unbounded region or of an unbounded function

- iterated integral
- for a function $f(x,y)$ over the region $R$ is
- ${\int}_{a}^{b}{\displaystyle {\int}_{c}^{d}f(x,y)dx\phantom{\rule{0.2em}{0ex}}dy}}={\displaystyle {\int}_{a}^{b}\left[{\displaystyle {\int}_{c}^{d}f(x,y)dy}\right]}dx,$
- ${\int}_{c}^{d}{\displaystyle {\int}_{b}^{a}f(x,y)dx\phantom{\rule{0.2em}{0ex}}dy}}={\displaystyle {\int}_{c}^{d}\left[{\displaystyle {\int}_{a}^{b}f(x,y)dx}\right]}dy,$

- Jacobian
- the Jacobian $J\left(u,v\right)$ in two variables is a $2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinant:

$$J\left(u,v\right)=\left|\begin{array}{lll}\frac{\partial x}{\partial u}\hfill & & \frac{\partial y}{\partial u}\hfill \\ \frac{\partial x}{\partial v}\hfill & & \frac{\partial y}{\partial v}\hfill \end{array}\right|;$$

the Jacobian $J\left(u,v,w\right)$ in three variables is a $3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant:

$$J(u,v,w)=\left|\begin{array}{ccccc}\frac{\partial x}{\partial u}\hfill & & \frac{\partial y}{\partial u}\hfill & & \frac{\partial z}{\partial u}\hfill \\ \frac{\partial x}{\partial v}\hfill & & \frac{\partial y}{\partial v}\hfill & & \frac{\partial z}{\partial v}\hfill \\ \frac{\partial x}{\partial w}\hfill & & \frac{\partial y}{\partial w}\hfill & & \frac{\partial z}{\partial w}\hfill \end{array}\right|$$

- one-to-one transformation
- a transformation $T:G\to R$ defined as $T\left(u,v\right)=\left(x,y\right)$ is said to be one-to-one if no two points map to the same image point

- planar transformation
- a function $T$ that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

- polar rectangle
- the region enclosed between the circles $r=a$ and $r=b$ and the angles $\theta =\alpha $ and $\theta =\beta ;$ it is described as $R=\left\{\left(r,\theta \right)|a\le r\le b,\alpha \le \theta \le \beta \right\}$

- radius of gyration
- the distance from an object’s center of mass to its axis of rotation

- transformation
- a function that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

- triple integral
- the triple integral of a continuous function $f\left(x,y,z\right)$ over a rectangular solid box $B$ is the limit of a Riemann sum for a function of three variables, if this limit exists

- triple integral in cylindrical coordinates
- the limit of a triple Riemann sum, provided the following limit exists:

$$\underset{l,m,n\to \infty}{\text{lim}}{\displaystyle \sum _{i=1}^{l}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}f({r}_{ijk}^{*},{\theta}_{ijk}^{*},{z}_{ijk}^{*}){r}_{ijk}^{*}\text{\Delta}r\text{\Delta}\theta \text{\Delta}z}}}$$

- triple integral in spherical coordinates
- the limit of a triple Riemann sum, provided the following limit exists:

$$\underset{l,m,n\to \infty}{\text{lim}}{\displaystyle \sum _{i=1}^{l}{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{k=1}^{n}f({\rho}_{ijk}^{*},{\theta}_{ijk}^{*},{\phi}_{ijk}^{*}){({\rho}_{ijk}^{*})}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi \text{\Delta}\rho \text{\Delta}\theta \text{\Delta}\phi}}}$$

- Type I
- a region $D$ in the $xy$-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions ${g}_{1}\left(x\right)$ and ${g}_{2}\left(x\right)$

- Type II
- a region $D$ in the $xy$-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions ${h}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{h}_{2}\left(y\right)$