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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):

∬[a,b]×[c,d]f(x,y) dA=lim\substackmax Δxi→0max Δyj→0∑i=1m∑j=1nf(ui,vj)ΔxiΔyj, \iint \limits_{[a,b]×[c,d]} f(x,y) \ dA= \lim_{\substack{\text{max} \ \Delta {x_i} \to 0 \\ \text{max} \ \Delta {y_j} \to 0 }} \sum_{i=1}^m \sum_{j=1}^n f(u_i,v_j) \Delta x_i \Delta y_j,

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

∬Rf(x,y) dA=∬[a,b]×[c,d]g(x,y) dA \iint \limits_R f(x,y) \ dA= \iint \limits_{[a,b]×[c,d]} g(x,y) \ dA

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.

3. The double integral of the sum of two functions is equal to the sum of the integrals of these functions

∬R[f(x,y)+g(x,y)] dA=∬Rf(x,y) dA+∬Rg(x,y) dA \iint \limits_R [f(x,y)+g(x,y)] \ dA = \iint \limits_R f(x,y) \ dA + \iint \limits_R g(x,y) \ dA

4. The double integral of the difference of two functions is equal to the difference of the integrals of these functions:

∬R[f(x,y)–g(x,y)] dA=∬Rf(x,y) dA–∬Rg(x,y) dA \iint \limits_R [f(x,y)-g(x,y)] \ dA = \iint \limits_R f(x,y) \ dA – \iint \limits_R g(x,y) \ dA

5. A constant factor can be moved across the double integral sign:

∬Rkf(x,y) dA=k∬Rf(x,y) dA \iint \limits_R kf(x,y) \ dA = k \iint \limits_R f(x,y) \ dA

6. If f(x,y) ≤ g(x,y) in a region R, then

∬Rf(x,y) dA≤∬Rg(x,y) dA \iint \limits_R f(x,y) \ dA \le \iint \limits_R g(x,y) \ dA

7. If f(x,y) ≥ 0 in a region R and S⊂R, then

∬Sf(x,y) dA≤∬Rf(x,y) dA \iint \limits_S f(x,y) \ dA \le \iint \limits_R f(x,y) \ dA

8. If f(x,y) ≥ 0 in a region R and R and S are non-overlapping regions, then

∬R∪Sf(x,y) dA=∬Rf(x,y) dA+∬Sf(x,y) dA \iint \limits_{R\cup S} f(x,y) \ dA = \iint \limits_R f(x,y) \ dA + \iint \limits_S f(x,y) \ dA

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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∬Rf(x,y) dA=∫ab∫p(x)q(x)f(x,y) dydx \iint \limits_R f(x,y) \ dA = \int\limits_a^b \int\limits_{p(x)}^{q(x)} f(x,y) \ dy dx

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

∬Rf(x,y) dA=∫cd∫u(y)v(y)f(x,y) dxdy \iint \limits_R f(x,y) \ dA = \int\limits_c^d \int\limits_{u(y)}^{v(y)} f(x,y) \ dxdy

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

∬Rf(x,y) dA=∫ab∫cdf(x,y) dy dx=∫cd∫abf(x,y) dx dy \iint \limits_R f(x,y) \ dA = \int\limits_a^b \left( \int\limits_c^d f(x,y) \ dy \right) \ dx = \int\limits_c^d \left( \int\limits_a^b f(x,y) \ dx \right) \ dy

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

∬Rf(x,y) dxdy=∬Rg(x)h(y) dxdy=∫abg(x)dx∫cdh(y)dy \iint \limits_R f(x,y) \ dxdy = \iint \limits_R g(x)h(y) \ dxdy = \left( \int\limits_a^b g(x)dx \right)\left( \int\limits_c^d h(y)dy \right)

12. Change of variables:

∬Rf(x,y) dxdy=∬Sf[x(u,v),y(u,v)]∂(x,y)∂(u,v)dudv \iint \limits_R f(x,y) \ dxdy = \iint \limits_S f[x(u,v),y(u,v)] \left| \frac {\partial (x,y)}{\partial (u,v)} \right| dudv
$$ where, \ \left| \frac {\partial (x,y)}{\partial (u,v)} \right| = \left\vert \matrix{ \frac {\partial x}{\partial u} & \frac {\partial x}{\partial v} \cr \frac {\partial y}{\partial u} & \frac {\partial y}{\partial v} } \right\vert \neq 0 $$

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=rcosθ,y=rsinθ 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=∂(x,y)∂(r,θ)drdθ=rdrdθ dxdy= \left| \frac {\partial (x,y)}{\partial (r,\theta)}\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

∬Rf(x,y) dxdy=∫αβ∫g(θ)h(θ)f(rcosθ,rsinθ)rdrdθ \iint \limits_R f(x,y) \ dxdy = \int\limits_{\alpha}^{\beta} \int\limits_{g( \theta)}^{h( \theta)} f(r \cos \theta , r \sin \theta) rdrd \theta

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

∬Rf(x,y) dxdy=∫αβ∫abf(rcosθ,rsinθ)rdrdθ \iint \limits_R f(x,y) \ dxdy = \int\limits_{\alpha}^{\beta} \int\limits_a^b f(r \cos \theta , r \sin \theta) rdrd \theta

16. Area of a type I region

A=∫ab∫g(x)h(x)dydx A= \int\limits_a^b \int\limits_{g(x)}^{h(x)} dydx

17. Area of a type II region

A=∫cd∫p(y)q(y)dxdy A= \int\limits_c^d \int\limits_{p(y)}^{q(y)} dxdy

18. Volume of a solid

V=∬Rf(x,y) dA V= \iint \limits_R f(x,y) \ dA

If R is a type I region bounded by x=a, x=b, y=h(x), y=g(x), then

V=∬Rf(x,y) dA=∫ab∫h(x)g(x)f(x,y) dydx V= \iint \limits_R f(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=∬Rf(x,y) dA=∫cd∫p(y)q(y)f(x,y) dxdy V= \iint \limits_R f(x,y) \ dA = \int\limits_c^d \int\limits_{p(y)}^{q(y)} f(x,y) \ dxdy

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

V=∬R[f(x,y)–g(x,y)] dA V= \iint \limits_R [f(x,y)-g(x,y)] \ dA

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

A=∬S dA=∫αβ∫h(θ)g(θ)rdrdθ,V=∬Sf(r,θ)rdrdθ A = \iint \limits_S \ dA = \int\limits_{\alpha}^{\beta} \int\limits_{h(\theta)}^{g(\theta)} rdrd\theta, V= \iint \limits_S f(r,\theta) rdrd\theta

21. Surface area:

S=∬R1+∂z∂x2+∂z∂y2dxdy S= \iint \limits_R \sqrt {1+ \left ( \frac {\partial z}{\partial x} \right)^2 + \left ( \frac {\partial z}{\partial y} \right)^2 } dxdy

22. Mass of a lamina

m=∬Rρ(x,y) dA m= \iint \limits_R \rho (x,y) \ dA

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

Mx=∬Ryρ(x,y) dA M_x= \iint \limits_R y \rho (x,y) \ dA

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

My=∬Rxρ(x,y) dA M_y= \iint \limits_R x \rho (x,y) \ dA

24. Moments of inertia of a lamina

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

Ix=∬Ry2ρ(x,y) dA I_x= \iint \limits_R y^2 \rho (x,y) \ dA

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

Iy=∬Rx2ρ(x,y) dA I_y= \iint \limits_R x^2 \rho (x,y) \ dA

The polar moment of inertia is equal to

I0=∬R(x2+y2)ρ(x,y) dA I_0= \iint \limits_R (x^2+y^2) \rho (x,y) \ dA

25. Center of mass of a lamina

x¯=Mym=1m∬Rxρ(x,y) dA=∬Rxρ(x,y) dA∬Rρ(x,y) dA \bar x= \frac {M_y}{m} = \frac 1m \iint \limits_R x \rho (x,y) \ dA = \frac {\iint \limits_R x \rho (x,y) \ dA}{\iint \limits_R \rho (x,y) \ dA}
y¯=Mxm=1m∬Ryρ(x,y) dA=∬Ryρ(x,y) dA∬Rρ(x,y) dA \bar y= \frac {M_x}{m} = \frac 1m \iint \limits_R y \rho (x,y) \ dA = \frac {\iint \limits_R y \rho (x,y) \ dA}{\iint \limits_R \rho (x,y) \ dA}

26. Charge of a plate

Q=∬Rσ(x,y) dA Q= \iint \limits_R \sigma (x,y) \ dA

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

27. Average of a function

μ=1S∬Rf(x,y) dA, \mu = \frac 1S \iint \limits_R f(x,y) \ dA ,
where, S=∬R dA where, \ S= \iint \limits_R \ dA

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