Integration of functions of several real variables

Question 1

Single integrals

Answer 1

Let a function f(x) be defined in an interval [a,b] where a=x1 < x2 < … < xn+1 =b.
Then the integral of the function f(x) on the interval [a,b] is defined as follows
∫ (from a to b) f(x) .dx = lim (max |Δxi| -> 0) ∑ (from i=1,n) f(si) Δxi.
where Δxi = xi+1-xi

Question 2

Approximate area

Answer 2

f(si)Δxi is the approximate area of the subdomain below y=f(x) and above the segment Δxi

Question 3

Riemann sum (Single Integrals)

Answer 3

∑ (from i=1,n) f(si) Δxi, the Riemann sum, is the approximate area of the domain below y=f(x) and above the interval [a,b]

Question 4

Accurate area

Answer 4

∫ (from a to b) f(x) .dx = lim (max |Δxi| -> 0) ∑ (from i=1,n) f(si) Δxi, is the accurate area of the domain below y=f(x) and above the interval [a,b]

Question 5

Approximate volume

Answer 5

f(si, ti) Δxi Δyi is the approximate volume of the beam below the surface and above the rectangle with side Δxi Δyi.

Question 6

The Riemann sum (double integrals)

Answer 6

∑ (from i=1,n) ∑ (from j=1,m) f(si, tj) Δxi Δyi

is the approximate volume of a solid region S lying above the domain D in the x-y plane and below the surface z=f(x,y).

Question 7

Volume of a solid region S

Answer 7

The limit of the sum lim (max |Δxi| -> 0) lim (max |yxi| -> 0) ∑ (from i=1,n) ∑ (from j=1,m) f(si, tj) Δxi Δyi

Question 8

Double integral

Answer 8

The double integral of a continuous function f(x,y) defined on a closed domain D is defined by the following
∫∫ (over D) f(x,y) .dxdy = lim (max |Δxi| -> 0) (max |Δyi| -> 0)∑ (from i=1,n)∑ (from j=1,n) f(si,tj) Δxi Δyi.

where Δxi = xi+1 - xi Δyi = yi+1 - yi
si e [xi, xi+1], ti e [yi, yi+1]

Question 9

Fubinis theorem (for double integrals)

Answer 9

Let |R^2-> |R be continuous on D, which is given as
D = { (x,y) : a=< x=< b, c(x) =< y =< d(x) }
where c(x) and d(x) are smooth. If ∫∫ (over D) f(x,y) .dxdy then
∫∫ (over D) f(x,y) .dxdy = ∫ (over a, b)(∫ (over c(x), d(x) ) f(x,y).dy ) .dx
= ∫ (over a,b) .d(x) ∫(over c(x), d(x)) f(x,y) . dy

Question 10

Fubinis theorem (for triple integrals)

Answer 10

∫∫∫ (over T) f(x,y,z) .dxdydz
= ∫(over a , b) .dx ∫over c(x), d(x). dx ∫ (over l(x,y), h(x,y) f(x,y,z).dz

where T ={ (x,y,z) : a=< x =< b, c(x) =

Question 11

Symmetric properties of integrals

Answer 11

If S is symmetric to the plane x=0 and f(x,y,z) is odd in x, e.g. f(-x, y, z) = -f(x, y, z) then
∫∫∫ (over S) f(x,y,z) =0

If S is symmetric to the plane x=0 and f(x,y,z) is even in x (e.g. f(-x, y, z) = f(x, y, z) then
∫∫∫ (over S) f(x,y,z) = 2 ∫∫∫ (over S+) f(x,y,z)

Question 12

Polar coordinates

Answer 12

y = rcosθ
x= rsinθ

where r = √x^2 + y^2 and tanθ = y/x

Question 13

Double integral in polar coordinates

Answer 13

∫∫ (over D) f(x,y) . dxdy = ∫∫ (over D) f( rcosθ, rsinθ) rdrdθ

Question 14

Domain of polar coordinates

Answer 14

R = { θmin =< θ =< θmax, r1 (θ) =< r =< r2(θ)}

Question 15

The Jacobian (of double integrals)

Answer 15

The Jacobian of the functions x(u,v) and y(u,v) with respect to the variables u and v is the determinant
J = ∂(x,y) / ∂(u,v) = |xu xv|
|yu yv|

Question 16

Changing variables of double integrals

Answer 16

dA = dxdy = ∂(x,y) / ∂(u,v) dudv

∫∫ (over D) f(x,y) dxdy = ∫∫ (over D’) f(x,y) ∂(x,y) / ∂(u,v) dudv

where D’ is the region of the (u,v)-plane corresponding to the region of D of the (x,y)-plane.

Question 17

Cylindrical coordinates

Answer 17

Consider a point P with cartesian coordinates (x,y,z).
Let Q be the projection of P on the 0xy plane.
Denote its cylindrical coordinates as (r, θ, z), then
{ x = rcosθ <=> {r = √x^2 + y^2
y= rsinθ tanθ = y/x
z= z z=z

Question 18

Coordinate surfaces of cylindrical coordinates

Answer 18

r = r0 is a cylinder of radius r0, centred along the -axis
θ = θ0 is a half plane emanating from the z-axis
z=z0 is a plane parallel to the x-y plane.

Question 19

The Jacobian of triple integrals

Answer 19

The Jacobian of the functions x(u, v ,w), y(u, v, w) and z(u, v, w) w.r.t the variables u, v and w is the determinant
J= ∂(x,y,z) / ∂(u,v,w) = |xu xv xw|
|yu yv yw|
|zu, zv, zw|

Question 20

Changing variables of triple integrals

Answer 20

dV = dxdydz = ∂(x,y,z) / ∂(u,v,w) dudvdw

∫∫ (over S) f(x,y,z) dxdydz = ∫∫ (over S’) f(x(u,v,w) , y(u,v,w), z(u,v,w)) ∂(x,y,z) / ∂(u,v,w) dudvdw

where S’ is the region of the (u,v,w)-space corresponding to the region of S of the (x,y,z)-space

Question 21

Triple integrals in cylindrical coordinates

Answer 21

For the cylindrical coordinates, we have x=rcosθ , y=rsinθ , z=z.

dV = dxdydz = rdrdθdz

Question 22

Spherical coordinates (ρ, ѱ, θ)

Answer 22

x = ρsinѱcosθ
y= ρsinѱcosθ
z= ρcosѱ

where ρ = √x^2 + y^2 + z^2
θ = arctan(y/x)
ѱ = arccos(z/ρ)

Question 23

Coordinate surfaces of spherical coordinates

Answer 23

ρ = ρ0 is a sphere of radius ρ0, centred at the origin
ѱ=ѱ0, aziu angle, a circular cone whose vertex is at the origin
θ=θ0, colatitude angle, a half-plane emanating from the z-axis

Question 24

Triple integrals in spherical coordinates

Answer 24

dV = J dρdѱdθ= ρ^2sinѱdρdѱdθ

∫∫∫ (over S) f(x,y,z) dxdydz = ∫∫∫ (over S’) f(ρ,ѱ,θ)ρ^2 dρdѱdθ