Key Equations

Key Equations

Double integral	${\displaystyle \underset{R}{\iint }f(x,y)dA}=\underset{m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n}f(x_{ij}^{*},y_{ij}^{*})\text{Δ}A}}$
Iterated integral	${\displaystyle {\int }_a^b{\displaystyle {\int }_c^{d}f(x,y)dxdy}}={\displaystyle {\int }_a^b\left[{\displaystyle {\int }_c^{d}f(x,y)dy}\right]}dx$ or ${\displaystyle {\int }_c^d{\displaystyle {\int }_b^{a}f(x,y)dxdy}}={\displaystyle {\int }_c^d\left[{\displaystyle {\int }_a^{b}f(x,y)dx}\right]}dy$
Average value of a function of two variables	$f_{\text{ave}}=\frac{1}{\text{Area}R}{\displaystyle \underset{R}{\iint }f(x,y)dxdy}$

Iterated integral over a Type I region	${\displaystyle \underset{D}{\iint }f(x,y)dA}={\displaystyle \underset{D}{\iint }f(x,y)dydx}={\displaystyle \underset{a}{\overset{b}{\int }}\left[{\displaystyle \underset{g_1(x)}{\overset{g_2(x)}{\int }}f(x,y)dy}\right]dx}$
Iterated integral over a Type II region	${\displaystyle \underset{D}{\iint }f(x,y)dA}={\displaystyle \underset{D}{\iint }f(x,y)dxdy}={\displaystyle \underset{c}{\overset{d}{\int }}\left[{\displaystyle \underset{h_1(y)}{\overset{h_2(y)}{\int }}f(x,y)dx}\right]dy}$

Double integral over a polar rectangular region $R$	${\displaystyle \underset{R}{\iint }f\left(r,\theta \right)}dA=\underset{m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n}f\left(r_{ij}*,{\theta }_{ij}*\right)}}\Delta A=\underset{m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^{n}f\left(r_{ij}*,{\theta }_{ij}*\right)}}{r}_{ij}*\Delta r\Delta \theta$
Double integral over a general polar region	${\displaystyle \underset{D}{\iint }f\left(r,\theta \right)}rdrd\theta ={\displaystyle \underset{\theta =\alpha }{\overset{\theta =\beta }{\int }}{\displaystyle \underset{r=h_1\left(\theta \right)}{\overset{r=h_2\left(\theta \right)}{\int }}f\left(r,\theta \right)}}rdrd\theta$

Triple integral

$\underset{l,m,n\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^{n}f(x_{ijk}^{*},y_{ijk}^{*},z_{ijk}^{*})\text{Δ}x\text{Δ}y\text{Δ}z}}}={\displaystyle \underset{B}{\iiint }f(x,y,z)dV}$

Triple integral in cylindrical coordinates	${\displaystyle \underset{B}{\iiint }g\left(x,y,z\right)dV}={\displaystyle \underset{B}{\iiint }g\left(r\text{cos}\theta ,r\text{sin}\theta ,z\right)rdrd\theta dz={\displaystyle \underset{B}{\iiint }f\left(r,\theta ,z\right)}}rdrd\theta dz$
Triple integral in spherical coordinates	${\displaystyle \underset{B}{\iiint }f\left(\rho ,\theta ,\phi \right)}{\rho }^2\text{sin}\phi d\rho d\phi d\theta ={\displaystyle \underset{\phi =\gamma }{\overset{\phi =\psi }{\int }}{\displaystyle \underset{\theta =\alpha }{\overset{\theta =\beta }{\int }}{\displaystyle \underset{\rho =a}{\overset{\rho =b}{\int }}f\left(\rho ,\theta ,\phi \right)}}}{\rho }^2\text{sin}\phi d\rho d\phi d\theta$

Mass of a lamina	$m=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l{m}_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }\rho (x,y)dA}}}$
Moment about the x-axis	$M_x=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(y_{ij}^{*}\right)m_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(y_{ij}^{*}\right)\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }y\rho (x,y)dA}}}$
Moment about the y-axis	$M_y=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(x_{ij}^{*}\right)m_{ij}}}=\underset{k,l\to \infty }{\text{lim}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^l\left(x_{ij}^{*}\right)\rho (x_{ij}^{*},y_{ij}^{*})\text{Δ}A={\displaystyle \underset{R}{\iint }x\rho (x,y)dA}}}$
Center of mass of a lamina	$\stackrel{\text{−}}{x}=\frac{M_y}{m}=\frac{{\displaystyle \underset{R}{\iint }x\rho \left(x,y\right)dA}}{{\displaystyle \underset{R}{\iint }\rho \left(x,y\right)dA}}$ and $\stackrel{\text{−}}{y}=\frac{M_x}{m}=\frac{{\displaystyle \underset{R}{\iint }y\rho \left(x,y\right)dA}}{{\displaystyle \underset{R}{\iint }\rho \left(x,y\right)dA}}$