# Integral calculus - Double integral

Functions of two variables : f(x,y), f(u,v),....
Double integrals : $\iint\limits_Rf(x,y)dxdy, \iint\limits_Rg(x,y)dxdy, ...$

Riemann sum : $\displaystyle\sum_{i=1}^m \sum_{j=1}^n f \bigg(u_i,v_i \bigg) \bigtriangleup x_i \bigtriangleup y_j$
Small Changes :$\bigtriangleup x_i \bigtriangleup y_j$
Regions of integration : R, S
Polar Coordinates :r, $\theta$
Area : A
Surface area : S
Volume of a solid : V
Mass of a lamina : m
Density : p(x,y)
First moments of inertia : $I_x,I_y,I_0$
Charge of a plate : Q
Charge density : $\sigma(x,y)$
Coordinates of center of mass : $\overline x, \overline y$
Average of a function : $\mu$

1. Definition of Double Integral
The double integral over a rectangle [a,b] $\times$[c,d] is defined to be
$\displaystyle\iint\limits_{[a,b] \times [c,d]} f(x,y)dA=\lim_{\substack{max \bigtriangleup x_i\to0\\max \bigtriangleup x_j\to0}} \sum_{i=1}^m \sum_{j=1}^n f \bigg (u_i, v_j \bigg ) \bigtriangleup x_i \bigtriangleup y_j$
Where ($u_i,v_j$) is some point in the rectangle
$(x_{i-1}, x_i) \times ( y_{j-1}, y_j), and \; \bigtriangleup x_i=x_i-x_{i-1}, \bigtriangleup y_j=y_j-y_{j-1}.$ The double integral over a general region R is
$\iint\limits_R f(x,y)dA= \iint \limits_ {[a,b] \times [c,d]} g(x,y)dA,$
Where rectangle $[a,b] \times [c,d]$ contains R,
g(x,y)=f(xx,y) if f(xx,y) is in R and g(x,y)=0 Otherwise. 2. $\iint\limits_R[f(x,y)+g(x,y)]dA=\iint\limits_Rf(x,y)dA+\iint\limits_Rg(x,y)dA$
3. $\iint\limits_R[f(x,y)-g(x,y)]dA=\iint\limits_Rf(x,y)dA-\iint\limits_Rg(x,y)dA$
4. $\iint\limits_Rkf(xy)dA= k\iint\limits_Rf(x,y)dA,$
Where k i s a constant
5. if f(x,y) $\leq$ g(x,y) on R, then $\iint\limits_Rf(x,y)dA \leq \iint\limits_Rg(x,y)dA.$
6. iff f(x,y) $\geq0$ on R and  $S \subset R$, then
$\iint\limits_sf(x,y)dA \leq \iint\limits_Rf(x,y)dA.$ 7. if f(x,y)$\geq$0 om R and R and S are non- overlapping regions , then $\iint \limits_{R\, \cup \,S}f(x,y)dA= \iint \limits_Rf(x,y)dA+ \iint \limits_Sf(x,y)dA .$
Here $R \cup S$ is the union of the regions R and S. 8. Iterated Integrals and Fubini's Theorem
$\iint \limits_Rf(xx,y)dA= \int\limits_a^b \int\limits_{p(x)}^{q(x)}f(x,y)dy\, dx$
for a region of type I,
R={(x,y)|a$\leq$x$\leq$b, p(x)$\leq$y$\leq$q(x) }. $\iint \limits_Rf(x,y)dA= \int\limits_c^d \int\limits_{u(y)}^{v(y)} f(x,y)dx\,dy$
for a region of type II,
R={(x,y)|u(y)$\leq$$\leq$v(Y) , c $\leq$ y $\leq$d }. 9. Double Integrals over Rectangular Regions
if R is the rectangular region [a,b] $\times$[c,d], then
$\iint\limits_Rf(x,y)dx\,dy= \int\limits_a^b\bigg ( {\int\limits_c^df(x,y)dy} \bigg)dx=\int\limits_c^d \bigg ( {\int\limits_a^b f(x,y)dx} \bigg)dy.$
In the special case where the integrand f (x,y) can be written as g(x)h(y) we have
$\iint\limits_Rf(x,y)dx \, dy= \iint\limits_R g(x)h(y)dx\,dy= \bigg ( { \int\limits_a^b g(x)dx} \bigg ) \bigg ( {\int\limits_c^d h(y)dy }\bigg).$
10. Change of Variables
$\iint\limits_Rf(x,y)dxdy= \iint\limits_Sf[x(u,v),y(u,v)] \bigg|{\partial(x,y) \over \partial(u,v) }\bigg|dudv,$
Where $\bigg|{\partial(x,y) \over \partial(u,v) }\bigg| = \Bigg | \substack{{\partial x \over \partial u } \ \ {\partial x \over \partial v } \\ {\partial x \over \partial u } \ \ {\partial x \over \partial v } } \Bigg | \neq 0$ is the jacobian of the transformations $(x,y) \to (u,v),$and S is the pullback of R which can be computed by $x=x(u,v), y=y(u,v)$ into the definition of R.
11. Polar Coordinates
$x=r\,cos \theta, \, y = r\, sin\theta.$ 12. Double Integrals in polar coordinates
The Differential dxdy for Polar Coordinates is
$dxdy= \bigg | {\partial(x,y) \over \partial(r,\theta)} \bigg |drd \, \theta =rdrd \, \theta.$
Let the region R is determined as follows :
$0\neq g(\theta)\leq r \leq h(\theta), \alpha \leq \theta \leq \beta , Where \;\; \beta-\alpha \leq 2 \pi.$
Then
$\iint\limits_R f(x,y)dxddy =\int\limits_\alpha^\beta \int\limits_{g(\theta)}^{h(\theta)} f (r\; cos\theta,r\,sin\theta)rdrd\,\theta \;.$ If the region R is the polar rectangle given by
$0 \le a \le r \le b, \alpha \le \theta \le \beta,$ where  $\beta - \alpha \le 2 \pi ,$
then
$\iint\limits_R f(x,y) dxdy= \int\limits_R^\beta \int\limits_a^b f(r\; cos \theta,r \; sin \theta ) rdrd \theta .$ 13. Area of a Region
$A= \int \limits_c^d \int\limits_{p(y)}^{q(y)} dx\,dy$ ( for a type II region). 14. Volume of a solid
$V= \iint \limits_R f(x,y) dA.$ If R is a type II region bounded  by x = a , x = b , y = h(x) ,y = g(x) , then
$V = \iint\limits_Rf(x,y)dA=\int\limits_a^b \int \limits_{h(x)}^{g(x)} f(x,y)dydx.$
If R is a type II region bounded  by y = c, y = d , x = q(y) , x = p(y) ,  then
$V = \iint\limits_Rf(x,y)dA=\int\limits_c^d \int \limits_{p(x)}^{q(x)} f(x,y)dydx.$
If $f(x,y) \ge g(x,y)$ over a region R, then the volume of the solid between $z_1=f(x,y)$ and $z_2=g(x,y)$ over R is given by
$V= \iint \limits_R[f(x,y)-g(x,y)]dA$.