Key Equations

• Double integral
  $\underset{R}{\displaystyle\iint} f(x,y)dA=\underset{m,n\to{\infty}}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}$
• 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
  $f_{\text{ave}}=\frac{1}{\text{Area R}}\underset{R}{\displaystyle\iint} f(x,y)dxdy$

Glossary

double Riemann Sum

of the function $f(x,y)$ over a rectangular region $R$ is $\displaystyle\sum_{i=1}^{m} {} \displaystyle\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}$ where $R$ is divided into smaller sub rectangles $R_{ij}$ and $({x^{*}}_{i,j}, {y^{*}}_{i,j})$ is an arbitrary point in $R_{ij}$

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, $\underset{R}{\displaystyle\iint} f(x,y)dA=\underset{m,n\to{\infty}}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}$

Fubini’s Theorem

if $f(x,y)$ is a function of two variables that is continuous over a rectangular region $R = \{(x,y)\in{\mathbb{R}}^{2}|a\leq x\leq b,c\leq y\leq d\}$, then the double integral of $f$ over the region equals an iterated integral,

$\underset{R}{\displaystyle\iint} f(x,y)dxdy={\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}{\displaystyle\int_{a}^{b} {f(x,y){dx}{dy}}}$

iterated Integral

for a function $f(x,y)$ over the region $\bf{R}$ is

${\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{a}^{b}}\left[{\displaystyle\int_{c}^{d} {f(x,y){dy}}}\right]{dx}$

${\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}\left[{\displaystyle\int_{a}^{b} {f(x,y){dx}}}\right]{dy}$