# Properties and Applications of Double Integrals

Functions of two variables: $$f\left( {x,y} \right)$$, $$f\left( {u,v} \right)$$, $$g\left( {x,y} \right)$$
Independent variables: $$x$$, $$y$$, $$u$$, $$v$$
Small changes: $$\Delta {x_i}$$, $$\Delta {y_i}$$
Regions of integration: $$R$$, $$S$$
Real numbers: $$a$$, $$b$$, $$c$$, $$d,$$ $$\alpha,$$ $$\beta$$
Polar coordinates: $$r$$, $$\theta$$
Area of a region: $$A$$
Surface area: $$S$$
Volume of a solid: $$V$$
Mass of a lamina: $$m$$
Density of a lamina: $$\rho\left( {x,y} \right)$$
First moments: $${M_x}$$, $${M_y}$$
Moments of inertia: $${I_x}$$, $${I_y}$$, $${I_0}$$
Charge of a plate: $$Q$$
Charge density: $$\sigma\left( {x,y} \right)$$
Coordinates of the center of mass: $$\bar x$$, $$\bar y$$
Average of a function: $$\mu$$
1. The double integral of a function $$f\left( {x,y} \right)$$ over a rectangular region $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ is defined as the limit of the integral sum (Riemann sum):
$$\require{AMSmath.js}{\large\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]}\normalsize} {f\left( {x,y} \right)dA} =$$ $$\lim\limits_{\substack{ \text{max}\,\Delta {x_i} \to 0\\ \text{max}\,\Delta {y_j} \to 0}} \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {f\left( {{u_i},{v_j}} \right)\Delta {x_i}\Delta {y_j}} },$$
where $${\left( {{u_i},{v_j}} \right)}$$ is some point in the rectangle $$\left( {{x_{i – 1}},{x_i}} \right) \times \left( {{y_{j – 1}},{y_j}} \right)$$ and $$\Delta {x_i} = {x_i} – {x_{i – 1}},$$ $$\Delta {y_j} = {y_j} – {y_{j – 1}}.$$
2. The double integral of a function $$f\left( {x,y} \right)$$ over a general region $$R$$ is defined to be
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\iint\limits_{\left[ {a,b} \right] \times \left[ {c,d} \right]}\normalsize} {g\left( {x,y} \right)dA},$$
where the rectangle $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$ contains the region $$R$$. The function $$g\left( {x,y} \right)$$ is defined as follows:
$$g\left( {x,y} \right) = f\left( {x,y} \right)$$ when $$f\left( {x,y} \right)$$ is in $$R$$ and $$g\left( {x,y} \right) = 0$$ otherwise.
3. The double integral of the sum of two functions is equal to the sum of the integrals of these functions
$${\large\iint\limits_R\normalsize} {\left[ {f\left( {x,y} \right) + g\left( {x,y} \right)} \right]dA} =$$ $${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$ $$+\;{\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA}$$
4. The double integral of the difference of two functions is equal to the difference of the integrals of these functions:
$${\large\iint\limits_R\normalsize} {\left[ {f\left( {x,y} \right) – g\left( {x,y} \right)} \right]dA} =$$ $${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$ $$-\;{\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA}$$
5. A constant factor can be moved across the double integral sign:
$${\large\iint\limits_R\normalsize} {kf\left( {x,y} \right)dA} =$$ $$k{\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$
6. If $$f\left( {x,y} \right) \le g\left( {x,y} \right)$$ in a region $$R$$, then
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} \le$$ $${\large\iint\limits_R\normalsize} {g\left( {x,y} \right)dA}$$
7. If $$f\left( {x,y} \right) \ge 0$$ in a region $$R$$ and $$S \subset R$$, then
$${\large\iint\limits_S\normalsize} {f\left( {x,y} \right)dA} \le$$ $${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$
8. If $$f\left( {x,y} \right) \ge 0$$ in a region $$R$$ and $$R$$ and $$S$$ are non-overlapping regions, then
$${\large\iint\limits_{R \cup S}\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$ $$+\;{\large\iint\limits_S\normalsize} {f\left( {x,y} \right)dA}.$$
Here $${R \cup S}$$ is the union of the regions of integration $$R$$ and $$S.$$
9. Iterated integrals and Fubini’s theorem for a region of type $$I$$
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\int\limits_a^b\normalsize} {{\large\int\limits_{p\left( x \right)}^{q\left( x \right)}\normalsize} {f\left( {x,y} \right)dy\,dx} },$$
where the region of integration $$R$$ is defined by the inequalities
$$R = \big\{ \left( {x,y} \right) \mid a \le x \le b,$$ $$p \left( x \right) \le y \le q\left( x \right)\big\}.$$
10. Iterated integrals and Fubini’s theorem for a region of type $$II$$
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\int\limits_c^d\normalsize} {{\large\int\limits_{u\left( y \right)}^{v\left( y \right)}\normalsize} {f\left( {x,y} \right)dx\,dy} },$$
where the region of integration $$R$$ is defined by the inequalities
$$R = \big\{ \left( {x,y} \right) \mid u\left( y \right) \le x \le v\left( y \right),$$ $$c \le y \le d \big\}.$$
11. Double integrals over rectangular regions
If $$R$$ is a rectangular region $$\left[ {a,b} \right] \times \left[ {c,d} \right]$$, then
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\int\limits_a^b\normalsize} {\left( {{\large\int\limits_c^d\normalsize} {f\left( {x,y} \right)dy} } \right)dx} =$$ $${\large\int\limits_c^d\normalsize} {\left( {\large\int\limits_a^b\normalsize {f\left( {x,y} \right)dx} } \right)dy}$$
In the special case when the integrand $$f\left( {x,y} \right)$$ can be written as the product $$g\left( x \right) h\left( y \right)$$, the double integral is given by
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =$$ $${\large\iint\limits_R\normalsize} {g\left( x \right)h\left( y \right)dxdy} =$$ $$\left( {{\large\int\limits_a^b\normalsize} {g\left( x \right)dx} } \right)\left( {{\large\int\limits_c^d\normalsize} {h\left( y \right)dy} } \right)$$
12. Change of variables
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =$$ $${\large\iint\limits_S\normalsize} {f\left[ {x\left( {u,v} \right),y\left( {u,v} \right)} \right] }$$ $${ \left| {\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}}\normalsize} \right|dudv},$$
where $$\left|{\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}}\normalsize}\right| =$$ $$\left| {\begin{array}{*{20}{c}} {\large\frac{{\partial x}}{{\partial u}}\normalsize} & {\large\frac{{\partial x}}{{\partial v}}\normalsize}\\ {\large\frac{{\partial y}}{{\partial u}}\normalsize} & {\large\frac{{\partial y}}{{\partial v}}\normalsize} \end{array}} \right|$$ $$\ne 0$$ is the jacobian of the transformation $$\left( {x,y} \right) \to \left( {u,v} \right)$$ and $$S$$ is the pullback of the region $$R$$ which can be computed by substituting $$x = x\left( {u,v} \right)$$, $$y = y\left( {u,v} \right)$$ into the definition of $$R$$.
13. Polar coordinates
$$x = r\cos \theta,$$ $$y = r\sin \theta$$
14. Double integrals in polar coordinates
The differential $$dxdy$$ in polar coordinates is given by the expression
$$dxdy = \left| {\large\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {r,\theta } \right)}}\normalsize} \right|drd\theta =$$ $$rdrd\theta$$
Let the region of integration $$R$$ be defined as follows:
$$0 \le g\left( \theta \right) \le r \le h\left( \theta \right)$$, $$\alpha \le \theta \le \beta,$$
where $$\beta – \alpha \le 2\pi$$. Then
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =$$ $${\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_{g\left( \theta \right)}^{h\left( \theta \right)}\normalsize} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } }$$
15. Double integral over a polar rectangle
If the region of integration $$R$$ is a polar rectangle given by the inequalities $$0 \le a \le r \le b$$, $$\alpha \le \theta \le \beta,$$ where $$\beta – \alpha \le 2\pi$$, then
$${\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dxdy} =$$ $${\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_a^b\normalsize} {f\left( {r\cos \theta ,r\sin \theta } \right)rdrd\theta } }$$
16. Area of a type $$I$$ region
$$A = {\large\int\limits_a^b\normalsize} {{\large\int\limits_{g\left( x \right)}^{h\left( x \right)}\normalsize} {dydx} }$$
17. Area of a type $$II$$ region
$$A = {\large\int\limits_c^d\normalsize} {{\large\int\limits_{p\left( y \right)}^{q\left( y \right)}\normalsize} {dxdy} }$$
18. Volume of a solid
$$V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA}$$
19. If $$R$$ is a type $$I$$ region bounded by $$x = a,$$ $$x = b,$$ $$y = h\left( x \right),$$ $$y = g\left( x \right),$$ then
$$V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\int\limits_a^b\normalsize} {{\large\int\limits_{h\left( x \right)}^{g\left( x \right)}\normalsize} {f\left( {x,y} \right)dydx} }$$
If $$R$$ is a type $$II$$ region bounded by $$y = c,$$ $$y = d,$$ $$x = q\left( y \right),$$ $$x = p\left( y \right),$$ then
$$V = {\large\iint\limits_R\normalsize} {f\left( {x,y} \right)dA} =$$ $${\large\int\limits_c^d\normalsize} {{\large\int\limits_{p\left( y \right)}^{q\left( y \right)}\normalsize} {f\left( {x,y} \right)dxdy} }$$
20. Volume of a solid between two surfaces
If $$f\left( {x,y} \right) \ge g\left( {x,y} \right)$$ over a region $$R$$, then the volume of the solid between the surfaces $${z_1}\left( {x,y} \right)$$ and $${z_2}\left( {x,y} \right)$$ over $$R$$ is given by
$$V = {\large\iint\limits_R\normalsize} {\big[ {f\left( {x,y} \right) }}$$ $$-\;{{ g\left( {x,y} \right)} \big]dA}$$
21. Area and volume in polar coordinates
Let a region $$S$$ be given in polar coordinates in the $$xy$$-plane and bounded by the lines $$\theta = \alpha,$$ $$\theta = \beta,$$ $$r = h\left( \theta \right),$$ $$r = g\left( \theta \right).$$ Let also a function $$f\left( {r,\theta} \right)$$ be given in the region $$S$$. Then the area of the region $$S$$ and volume of the solid bounded by the surface $$f\left( {r,\theta} \right)$$ are determined by the formulas
$$A = {\large\iint\limits_S\normalsize} {dA} =$$ $${\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_{h\left( \theta \right)}^{g\left( \theta \right)}\normalsize} {rdrd\theta } } ,$$ $$V = {\large\iint\limits_S\normalsize} {f\left( {r,\theta } \right)rdrd\theta }$$
22. Surface area
$$S =$$ $${\large\iint\limits_R\normalsize} {\sqrt {1 + {{\left( {\large\frac{{\partial z}}{{\partial x}}\normalsize} \right)}^2} + {{\left( {\large\frac{{\partial z}}{{\partial y}}\normalsize} \right)}^2}} dxdy}$$
23. Mass of a lamina
$$m = {\large\iint\limits_R\normalsize} {\rho \left( {x,y} \right)dA},$$
where the lamina occupies the region $$R$$ and its density at a point $${\left( {x,y} \right)}$$ is $${\rho \left( {x,y} \right)}.$$
24. Static moments of a lamina
The static moment of a lamina about the $$x$$-axis is given by the formula
$${M_x} = {\large\iint\limits_R\normalsize} {y\rho \left( {x,y} \right)dA}$$
Similarly, the static moment of a lamina about the $$y$$-axis is expressed in the form
$${M_y} = {\large\iint\limits_R\normalsize} {x\rho \left( {x,y} \right)dA}$$
25. Moments of inertia of a lamina
The moment of inertia about the $$x$$-axis is given by
$${I_x} = {\large\iint\limits_R\normalsize} {{y^2}\rho \left( {x,y} \right)dA}$$
The moment of inertia about the $$y$$-axis is determined by the formula
$${I_y} = {\large\iint\limits_R\normalsize} {{x^2}\rho \left( {x,y} \right)dA}$$
The polar moment of inertia is equal to
$${I_0} = {\large\iint\limits_R\normalsize} {\left({x^2} + {y^2}\right) }$$ $${\rho \left( {x,y} \right)dA}$$
26. Center of mass of a lamina
$$\bar x = {\large\frac{{{M_y}}}{m}\normalsize} =$$ $${\large\frac{1}{m}\normalsize} {\large\iint\limits_R\normalsize} {x\rho \left( {x,y} \right)dA} =$$ $${\large\frac{{\iint\limits_R {x\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}\normalsize,}\;$$
$$\bar y = {\large\frac{{{M_x}}}{m}\normalsize} =$$ $${\large\frac{1}{m}\normalsize} {\large\iint\limits_R\normalsize} {y\rho \left( {x,y} \right)dA} =$$ $${\large\frac{{\iint\limits_R {y\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}\normalsize} .$$
27. Charge of a plate
$$Q = {\large\iint\limits_R\normalsize} {\sigma \left( {x,y} \right)dA},$$
where the electrical charge is distributed over the region $$R$$ and its density at a point $${\left( {x,y} \right)}$$ is $${\sigma \left( {x,y} \right)}.$$
28. Average of a function
$$\mu = {\large\frac{1}{S}\iint\limits_R\normalsize} {f\left( {x,y} \right)dA},$$
where $$S = {\large\iint\limits_R\normalsize} {dA}.$$