### Key Concepts

## 5.1 Double Integrals over Rectangular Regions

- We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.
- Properties of double integral are useful to simplify computation and find bounds on their values.
- We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral.
- Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.

## 5.2 Double Integrals over General Regions

- A general bounded region $D$ on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
- To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
- We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
- We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

## 5.3 Double Integrals in Polar Coordinates

- To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
- The area $dA$ in polar coordinates becomes $r\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta .$
- Use $x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ and $dA=r\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta $ to convert an integral in rectangular coordinates to an integral in polar coordinates.
- Use ${r}^{2}={x}^{2}+{y}^{2}$ and $\theta ={\text{tan}}^{\mathrm{-1}}\left(\frac{y}{x}\right)$ to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
- To find the volume in polar coordinates bounded above by a surface $z=f\left(r,\theta \right)$ over a region on the $xy$-plane, use a double integral in polar coordinates.

## 5.4 Triple Integrals

- To compute a triple integral we use Fubini’s theorem, which states that if $f\left(x,y,z\right)$ is continuous on a rectangular box $B=\left[a,b\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[c,d\right]\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left[e,f\right],$ then

$$\underset{B}{\iiint}f\left(x,y,z\right)dV={\displaystyle \underset{e}{\overset{f}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{c}{\overset{d}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{a}{\overset{b}{\int}}f\left(x,y,z\right)dx\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dz}}}$$

and is also equal to any of the other five possible orderings for the iterated triple integral. - To compute the volume of a general solid bounded region $E$ we use the triple integral

$$V\left(E\right)={\displaystyle \underset{E}{\iiint}1dV}.$$ - Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
- To compute the average value of a function over a general three-dimensional region, we use

$${f}_{\text{ave}}=\frac{1}{V\left(E\right)}{\displaystyle \underset{E}{\iiint}f\left(x,y,z\right)}dV.$$

## 5.5 Triple Integrals in Cylindrical and Spherical Coordinates

- To evaluate a triple integral in cylindrical coordinates, use the iterated integral

$$\underset{\theta =\alpha}{\overset{\theta =\beta}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{r={g}_{1}\left(\theta \right)}{\overset{r={g}_{2}\left(\theta \right)}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{z={u}_{1}\left(r,\theta \right)}{\overset{z={u}_{2}\left(r,\theta \right)}{\int}}f\left(r,\theta ,z\right)}}}r\phantom{\rule{0.2em}{0ex}}dz\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta .$$ - To evaluate a triple integral in spherical coordinates, use the iterated integral

$$\underset{\theta =\alpha}{\overset{\theta =\beta}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{\rho ={g}_{1}\left(\theta \right)}{\overset{\rho ={g}_{2}\left(\theta \right)}{\int}}\phantom{\rule{0.2em}{0ex}}{\displaystyle \underset{\phi ={u}_{1}\left(r,\theta \right)}{\overset{\phi ={u}_{2}\left(r,\theta \right)}{\int}}f\left(\rho ,\theta ,\phi \right)}}}{\rho}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}d\phi \phantom{\rule{0.2em}{0ex}}d\rho \phantom{\rule{0.2em}{0ex}}d\theta .$$

## 5.6 Calculating Centers of Mass and Moments of Inertia

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

- For a lamina $R$ with a density function $\rho \left(x,y\right)$ at any point $\left(x,y\right)$ in the plane, the mass is $m={\displaystyle \underset{R}{\iint}\rho \left(x,y\right)dA}.$
- The moments about the $x\text{-axis}$ and $y\text{-axis}$ are

$${M}_{x}={\displaystyle \underset{R}{\iint}y\rho \left(x,y\right)}dA\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{M}_{y}={\displaystyle \underset{R}{\iint}x\rho \left(x,y\right)}dA.$$ - The center of mass is given by $\stackrel{\text{\u2212}}{x}=\frac{{M}_{y}}{m},\stackrel{\text{\u2212}}{y}=\frac{{M}_{x}}{m}.$
- The center of mass becomes the centroid of the plane when the density is constant.
- The moments of inertia about the $x-\text{axis,}$ $y-\text{axis,}$ and the origin are

$${I}_{x}={\displaystyle \underset{R}{\iint}{y}^{2}\rho \left(x,y\right)dA,}\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{I}_{y}={\displaystyle \underset{R}{\iint}{x}^{2}\rho \left(x,y\right)dA,}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{0}={I}_{x}+{I}_{y}={\displaystyle \underset{R}{\iint}\left({x}^{2}+{y}^{2}\right)\rho \left(x,y\right)dA.}$$

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

- For a solid object $Q$ with a density function $\rho \left(x,y,z\right)$ at any point $\left(x,y,z\right)$ in space, the mass is $m={\displaystyle \underset{Q}{\iiint}\rho \left(x,y,z\right)dV}.$
- The moments about the $xy\text{-plane,}$ the $xz\text{-plane,}$ and the $yz\text{-plane}$ are

$${M}_{xy}={\displaystyle \underset{Q}{\iiint}z\rho \left(x,y,z\right)dV},\phantom{\rule{0.2em}{0ex}}{M}_{xz}={\displaystyle \underset{Q}{\iiint}y\rho \left(x,y,z\right)dV},\phantom{\rule{0.2em}{0ex}}{M}_{yz}={\displaystyle \underset{Q}{\iiint}x\rho \left(x,y,z\right)dV}.$$ - The center of mass is given by $\stackrel{\text{\u2212}}{x}=\frac{{M}_{yz}}{m},\stackrel{\text{\u2212}}{y}=\frac{{M}_{xz}}{m},\stackrel{\text{\u2212}}{z}=\frac{{M}_{xy}}{m}.$
- The center of mass becomes the centroid of the solid when the density is constant.
- The moments of inertia about the $yz\text{-plane,}$ the $xz\text{-plane,}$ and the $xy\text{-plane}$ are

$$\begin{array}{}\\ {I}_{x}={\displaystyle \underset{Q}{\iiint}\left({y}^{2}+{z}^{2}\right)\rho \left(x,y,z\right)dV,}{I}_{y}={\displaystyle \underset{Q}{\iiint}\left({x}^{2}+{z}^{2}\right)\rho \left(x,y,z\right)dV,}\hfill \\ {I}_{z}={\displaystyle \underset{Q}{\iiint}\left({x}^{2}+{y}^{2}\right)\rho \left(x,y,z\right)dV.}\hfill \end{array}$$

## 5.7 Change of Variables in Multiple Integrals

- A transformation $T$ is a function that transforms a region $G$ in one plane (space) into a region $R$ in another plane (space) by a change of variables.
- A transformation $T:G\to R$ defined as $T\left(u,v\right)=\left(x,y\right)$ $\left(\text{or}\phantom{\rule{0.2em}{0ex}}T\left(u,v,w\right)=\left(x,y,z\right)\right)$ is said to be a one-to-one transformation if no two points map to the same image point.
- If $f$ is continuous on $R,$ then $\underset{R}{\iint}f\left(x,y\right)}dA={\displaystyle \underset{S}{\iint}f\left(g\left(u,v\right),h\left(u,v\right)\right)}\left|\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}\right|du\phantom{\rule{0.2em}{0ex}}dv.$
- If $F$ is continuous on $R,$ then

$$\begin{array}{cc}\hfill {\displaystyle \underset{R}{\iiint}F\left(x,y,z\right)}dV& ={\displaystyle \underset{G}{\iiint}F\left(g\left(u,v,w\right),h\left(u,v,w\right),k\left(u,v,w\right)\right)}\left|\frac{\partial \left(x,y,z\right)}{\partial \left(u,v,w\right)}\right|du\phantom{\rule{0.2em}{0ex}}dv\phantom{\rule{0.2em}{0ex}}dw\hfill \\ & ={\displaystyle \underset{G}{\iiint}H\left(u,v,w\right)}\left|J\left(u,v,w\right)\right|du\phantom{\rule{0.2em}{0ex}}dv\phantom{\rule{0.2em}{0ex}}dw.\hfill \end{array}$$