## Double integration

### Double integration

Functions of two variables: f(x,y), f(u,v), g(x,y)

Independent variables: x, y, u, v

Small changes: Δxi, Δyi

Regions of integration: R, S

Real numbers: a, b, c, d, α, β

Polar coordinates: r, θ

Area of a region: A

Surface area: S

Volume of a solid: V

Mass of a lamina: m

Density of a lamina: ρ(x,y)

First moments: Mx, My

Moments of inertia: Ix, Iy, I0

Charge of a plate: Q

Charge density: σ(x,y)

Coordinates of the center of mass: $$\bar x$$, $$\bar y$$

Average of a function: μ

### 1. The double integral of a function f(x,y) over a rectangular region [a,b]×[c,d] is defined as the limit of the integral sum (Riemann sum):

where (ui,vj) is some point in the rectangle (xi−1,xi)×(yj−1,yj) and Δxi=xi−xi−1, Δyj=yj−yj−1. ### 2. The double integral of a function f(x,y) over a general region R is defined to be

where the rectangle [a,b]×[c,d] contains the region R. The function g(x,y) is defined as follows

g(x,y)=f(x,y) when f(x,y) is in R and g(x,y)=0 otherwise. ### 7. If f(x,y) ≥ 0 in a region R and S⊂R, then ### 8. If f(x,y) ≥ 0 in a region R and R and S are non-overlapping regions, then

Here R∪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

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where the region of integration R is defined by the inequalities

R={ (x,y) | a ≤ x ≤ b, p(x) ≤ y ≤ q(x)}. ### 10. Iterated integrals and Fubini’s theorem for a region of type II

where the region of integration R is defined by the inequalities

R={ (x,y) | u(y) ≤ x ≤ v(y), c ≤ y ≤ d}. ### 11. Double integrals over rectangular regions

If R is a rectangular region [a,b]×[c,d], then

In the special case when the integrand f(x,y) can be written as the product g(x)h(y), the double integral is given by

### 12. Change of variables:

$$where, \ \left| \frac {\partial (x,y)}{\partial (u,v)} \right| = \left\vert \matrix{ \frac {\partial x}{\partial u} & \frac {\partial x}{\partial v} \cr \frac {\partial y}{\partial u} & \frac {\partial y}{\partial v} } \right\vert \neq 0$$

This is the jacobian of the transformation (x,y)→(u,v) and S is the pullback of the region R which can be computed by substituting x=x(u,v), y=y(u,v) into the definition of R.

### 13. Polar coordinates

$x=r\mathrm{cos}\theta ,y=r\mathrm{sin}\theta$ ### 14. Double integrals in polar coordinates

The differential dxdy in polar coordinates is given by the expression

$dxdy=\left|\frac{\partial \left(x,y\right)}{\partial \left(r,\theta \right)}\right|drd\theta =rdrd\theta$

Let the region of integration R be defined as follows:

0 ≤ g(θ) ≤ r ≤ h(θ), α ≤ θ ≤ β,

where β − α ≤ 2π. Then ### 15. Double integral over a polar rectangle

If the region of integration R is a polar rectangle given by the inequalities 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β, where β − α ≤ 2π, then ### 16. Area of a type I region

$A=\underset{a}{\overset{b}{\int }}\underset{g\left(x\right)}{\overset{h\left(x\right)}{\int }}dydx$ ### 17. Area of a type II region

$A=\underset{c}{\overset{d}{\int }}\underset{p\left(y\right)}{\overset{q\left(y\right)}{\int }}dxdy$ ### 18. Volume of a solid If R is a type I region bounded by x=a, x=b, y=h(x), y=g(x), then

If R is a type II region bounded by y=c, y=d, x=q(y), x=p(y), then

### 19. Volume of a solid between two surfaces

If f(x,y) ≥ g(x,y) over a region R, then the volume of the solid between the surfaces z1(x,y) and z2(x,y) over R is given by

### 20. Area and volume in polar coordinates

Let a region S be given in polar coordinates in the xy-plane and bounded by the lines θ = α, θ = β, r = h(θ), r = g(θ). Let also a function f(r,θ) be given in the region S. Then the area of the region S and volume of the solid bounded by the surface f(r,θ) are determined by the formulas ### 21. Surface area:

$S=\underset{R}{\iint }\sqrt{1+{\left(\frac{\partial z}{\partial x}\right)}^{2}+{\left(\frac{\partial z}{\partial y}\right)}^{2}}dxdy$

### 22. Mass of a lamina

where the lamina occupies the region R and its density at a point (x,y) is ρ(x,y).

### 23. Static moments of a lamina

The static moment of a lamina about the x-axis is given by the formula

Similarly, the static moment of a lamina about the y-axis is expressed in the form

### 24. Moments of inertia of a lamina

The moment of inertia about the x-axis is given by

The moment of inertia about the y-axis is determined by the formula

The polar moment of inertia is equal to

### 26. Charge of a plate

where the electrical charge is distributed over the region R and its density at a point (x,y) is σ(x,y).