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Multiple Integrals

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Double integration

Representation of area as a double integral:

$A = lim↙{δx→0} ∑↙{x=a}↖{x=b} y . δx$

Expressed in integral notation as:
$A = ∫↙a↖b y dx$ or $∫↙a↖b f(x) dx$



$∬↙R$ dx dy represents the area of the region R


$∬↙R$ f(x, y) dx dy represents some physical quantity related to the area of region R.

Properties of the double integral

1. $∬↙R$ k f(x, y) dA = k $∬↙R$ f(x, y) dA

k is constant, free from x and y

2. $∬↙R$ [f(x, y) ± g(x, y)] dA = $∬↙R$ f(x, y) dA ± $∬↙R$ g(x, y) dA

3. if R = $R_1 $ ∪ $R_2$ and $R_1$ ∩ $R_2$ = ϕ then,

$∬↙R$ f(x, y) dA = $∬↙R_1$ f(x, y) dA + $∬↙R_2$ f(x, y) dA.


This result holds when R is the union of two non-overlapping regions $R_1$ and $R_2$

Evaluation of Double integrals


1. Suppose that R can be described by x = a, x = b, y = $f_1$(x) and y = $f_2$(x) then,

$I = ∬↙R f(x, y) dy dx = ∬↙a↖b \{ ∫↙{f_1 (x)}↖{f_2 (x)} f(x,y) dy \} dx$

2. Suppose that R can be described by y=c, y=d, x= $f_1$(y) and x = $f_2$ (y) then,


$I = ∬↙R f(x, y) dx dy = ∬↙c↖d \{ ∫↙{f_1 (y)}↖{f_2 (y)} f(x,y) dx \} dy$

3. Suppose that R can be described by x=a, x=b, y=c, and y=d (a rectangle) then,


$I = ∫↙a↖b \{ ∫↙c↖d f(x, y) dy \} dx$


or

$I = ∫↙c↖d \{ ∫↙a↖b f(x, y) dx \} dy$

4. Suppose R is same as mentioned in 3 above and the integrand is separable i.e. f(x, y) = u(x) . v(y), then

$∫↙{x=a}↖b ∫↙{y=c}↖d f(x, y) dy dx$ = $∫↙{x=a}↖b u(x) dx$ . $∫↙{y=c}↖d v(y) dy$

Determining the limits of integration

Method – I: Integrating first w.r.t. y and then w.r.t. x

Method – II: Integrating first w.r.t. x and then w.r.t. y

Type 3: Illustrations on change of order of integration

Method – I: Let the given integral be $I = ∫↙{x=a}↖{x=b} \{ ∫↙{y = f_1 (x)}↖{y= f_2 (x)} f(x, y) dy \} dx$

Method – II: Let the given integral be $I = ∫↙{y=c}↖{y=d} \{ ∫↙{x = g_1 (y)}↖{x= g_2 (y)} f(x, y) dx \} dy$

Transformation of cartesian double integral into polar double integral

$∫↙α↖β ∫↙0↖{f(θ)}$ r dθ represents the area OAB


$∫↙α↖β ∫↙0↖{f(θ)}$ F(r, θ) r dθ dr represents some physical quantity related to the area OAB

1. For a complete circle $x^2 + y^2 = a^2$

$I = ∫↙0↖{2π} \{ ∫↙0↖{a} f(r, θ) r dr\} dθ$

For a semi-circle $x^2 + y^2 = a^2$, y ≥ 0

$I = ∫↙0↖{π} \{ ∫↙0↖{a} f(r, θ) r dr\} dθ$

For a positive quadrant of a circle


$I = ∫↙0↖{π/2} \{ ∫↙0↖{a} f(r, θ) r dr\} dθ$


In all the three cases, polar equation of circle $x^2 + y^2 = a^2$ becomes r = a


2. For a circle $(x-a)^2 + (y-0)^2 = a^2$ having centre (a, 0), r = a
We write polar equation for $x^2 + y^2 = 2ax$ as $r^2$ = 2ar cosθ

$I = ∫↙{-π/2}↖{π/2} \{ ∫↙0↖{2acosθ} f(r, θ) r dr\} dθ$

For upper half of the circle


$I = ∫↙{0}↖{π/2} \{ ∫↙0↖{2acosθ} f(r, θ) r dr\} dθ$

3. For a circle $(x-0)^2 + (y-a)^2 = a^2$ having centre (0, a), r = a
We write polar equation for $x^2 + y^2 = 2ay$ as r = 2a sinθ

$I = ∫↙{0}↖{π} \{ ∫↙0↖{2asinθ} f(r, θ) r dr\} dθ$

4. For a cardiode r = a(1 + cosθ)

$I = ∫↙{0}↖{2π} \{ ∫↙0↖{a(1+cosθ)} f(r, θ) r dr\} dθ$

5. For $r^2 = a^2$ cos 2θ (Bernoullie’s Lemniscate)

$I = 2∫↙{-π/4}↖{π/4} \{ ∫↙0↖{a√{cos2θ}} f(r, θ) r dr\} dθ$ (for whole curve i.e. two loops)

$I = ∫↙{-π/4}↖{π/4} \{ ∫↙0↖{a√{cos2θ}} f(r, θ) r dr\} dθ$ (for one loop)

6. For a ellipse $x^2/a^2 + y^2/b^2 = 1$; Use x = ar cosθ, y = br sinθ, dx dy = ab r dr dθ


$I = ∫↙{0}↖{2π} \{ ∫↙0↖{1} f(r, θ)ab r dr\} dθ$


7. For a triangle y = 0, x = a, x = y

$I = ∫↙{0}↖{π/4} \{ ∫↙0↖{a secθ} f(r, θ) r dr\} dθ$

Type I: Direct evaluation

Type II: Evaluation by finding limits of integration

Triple Integration

Volume V of the tetrahedron is expressed as: $∫↙{x=0}↖{1} ∫↙{y=0}↖{1-x} ∫↙{z=0}↖{1-x-y} dz dy dx$

Mass M of tetrahedron: $∫↙{0}↖{1} ∫↙{0}↖{1-x} ∫↙{0}↖{1-x-y} f(x,y,z) dx dy dz$



1. Use of notation:

$∫↙{a}↖{b} ∫↙{f_1(x) ϕ_1}↖{f_2 (x) ϕ_2} ∫↙{(x,y)}↖{(x,y)} f(x,y,z) dz dy dx$ = $∫↙{a}↖{b}\{ ∫↙{f_1(x) ϕ_1}↖{f_2 (x) ϕ_2} \[ ∫↙{(x,y)}↖{(x,y)} f(x,y,z) dz \] dy \} dx$

Here solve inner integral first w.r.t. z then w.r.t. y and outermost integral finally w.r.t. x.

Note: The order of integration depends upon the distribution of limits

2. Use of spherical polar co-ordinates:



x = r sinθ cosϕ , y = r sinθ sinϕ , z = r cosθ , $x^2 + y^2 + z^2 = r^2 $ , ${δ (x,y,z)}/{δ(r,θ,ϕ)} = r^2 sinθ$


dx dy dz = $r^2$ sinθ dr dθ dϕ ; $I = ∫∫↙V∫ f(r,θ,ϕ) r^2 sinθ dr dθ dϕ$

Standard limits:

(i) For complete sphere $x^2 + y^2 + z^2 = a^2$, θ -> 0 to π, ϕ -> 0 to 2π, r -> 0 to a

(ii) For hemisphere $x^2 + y^2 + z^2 = a^2$, z > 0 : θ -> 0 to π/2, ϕ -> 0 to 2π, r -> 0 to a

(iii) For positive octant of a sphere $x^2 + y^2 + z^2 = a^2$ , θ -> 0 to π/2, ϕ -> 0 to π/2, r -> 0 to a

(iv) Also for ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$, we use

x = ar sinθ cosϕ , y = br sinθ sinϕ , z = cr cosθ , $x^2/a^2 + y^2/b^2 + z^2/c^2 = r^2 $ , $dx dy dz = abcr^2 sinθ dr dθ dϕ$


$I = ∫∫↙V∫ f(r,θ,ϕ) r^2 sinθ dr dθ dϕ$

Standard limits for ellipsoid: θ -> 0 to π, ϕ -> 2π , r -> 0 to 1

3. Use of cylindrical polar co-ordinates:

x = ρ cosϕ , y = ρ sinϕ , z = z, $x^2 + y^2 = ρ^2$, dx dy dz = ρ dρ dϕ dz, ${δ (x,y,z)}/{δ(ρ,ϕ,z)} = ρ$

$I = ∫∫↙V∫ f(ρ,ϕ,z) ρ dρ dϕ dz$

Standard limits: For cylinder $x^2 + y^2 + z^2 = a^2$, z = 0, z = h, ρ ->0 to a, ϕ -> 0 to 2π , z -> 0 to h