Engineering Mathematics II

Chapter 3
Multiple Integrals and Coordinate Transformations

Bachelor Program in Mechatronics Engineering
Engineering Mathematics II • Bachelor

Multiple Integrals and Coordinate Transformations

Introduction and Context

Multiple integration extends single-variable integration to quantities distributed over areas and volumes. In mechatronics engineering, it is used to determine mass, centres of mass, moments of inertia, fluid quantities, electric charge, stored energy, and probabilities over multidimensional regions. The central practical difficulty is often not integration itself, but describing the region correctly. Coordinate transformations—especially polar, cylindrical, and spherical coordinates—can convert complicated boundaries into simple ones.

A double integral accumulates a scalar field over a planar region, while a triple integral accumulates it through a solid. A Jacobian measures how an infinitesimal area or volume changes under a transformation. Correct limits, correct differential elements, and dimensional consistency are essential.

Learning Objectives

By the end of this chapter, you should be able to:

Interpret and evaluate double and triple integrals.
Set up iterated integrals over non-rectangular regions.
Change the order of integration when useful.
Use polar, cylindrical, and spherical coordinates appropriately.
Derive and apply Jacobian factors.
Calculate mass, centroids, moments of inertia, energy, and probability.
Assess modelling assumptions and numerical reliability in engineering applications.

1. Double Integrals in Cartesian Coordinates

For a continuous function f(x,y) over a bounded region R in the xy-plane, the double integral is defined as the limit of sums over small areas:

∬_R f(x,y) dA

Here, dA is an infinitesimal area element. If f ≥ 0, the integral represents volume beneath the surface z = f(x,y) and above R. More generally, it represents the accumulated value of a density or field.

For a rectangle R = [a,b] × [c,d]],

∬_R f(x,y) dA = ∫_a^b ∫_c^d f(x,y) dy dx

provided f is continuous on R. Reversing the order is also valid:

∫_a^b ∫_c^d f(x,y) dy dx = ∫_c^d ∫_a^b f(x,y) dx dy

This is a consequence of Fubini's theorem under standard continuity or integrability conditions.

Example 1: Rectangular Region

Evaluate ∫₀² ∫₁³ (x + 2y) dy dx.

First integrate with respect to y:

∫₁³ (x + 2y)dy = [xy + y²]₁³ = 2x + 8

Then integrate with respect to x:

∫₀² (2x + 8)dx = [x² + 8x]₀² = 20

Thus the value is 20.

2. Regions with Variable Boundaries

A vertically simple region can be written as

R = {(x,y): a ≤ x ≤ b, g₁(x) ≤ y ≤ g₂(x)}

and its integral is

∬_R f dA = ∫_a^b ∫_g₁(x)^g₂(x) f(x,y)dy dx

A horizontally simple region is described by

R = {(x,y): c ≤ y ≤ d, h₁(y) ≤ x ≤ h₂(y)}

with integral

∬_R f dA = ∫_c^d ∫_h₁(y)^h₂(y) f(x,y)dx dy

The region should always be sketched or mentally decomposed before limits are selected. If one set of slices requires splitting the region, the alternative order may be simpler.

Example 2: Triangular Region

Let R be bounded by x=0, y=0, and x+y=1. In vertical form,

0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x

Therefore,

∬_R (x+y)dA = ∫₀¹ ∫₀^1−x(x+y)dy dx

The inner integral is

[x y + y²/2]₀^1−x = x(1−x) + (1−x)²/2

which simplifies to (1−x²)/2. Hence,

∫₀¹ (1−x²)/2 dx = 1/3

3. Applications of Double Integrals

Area

The area of R is

A = ∬_R 1 dA

Mass of a Lamina

If a thin plate has surface density ρ(x,y) in units such as kg/m², then

m = ∬_R ρ(x,y)dA

Centroid

For a lamina of mass m, its centroid (x̄,ȳ) is

x̄ = (1/m) ∬_R xρ(x,y)dA,
ȳ = (1/m) ∬_R yρ(x,y)dA

The numerator has units of kg·m, so division by mass gives metres.

Moments of Inertia

About the x-axis and y-axis respectively,

I_x = ∬_R y²ρ(x,y)dA,
I_y = ∬_R x²ρ(x,y)dA

The polar moment about the origin is

I_O = I_x + I_y = ∬_R (x²+y²)ρ(x,y)dA

These formulas assume a thin lamina and density measured per unit area.

4. Polar Coordinates

Cartesian coordinates are inefficient for circular or radial geometry. Polar coordinates are defined by

x = r cosθ,
y = r sinθ,
r = √(x²+y²)

where r ≥ 0 is the distance from the origin and θ is the counterclockwise angle from the positive x-axis. The area element is

dA = r dr dθ

The factor r is the polar Jacobian. It must not be omitted.

A disk of radius a has limits 0 ≤ r ≤ a and 0 ≤ θ ≤ 2π. A sector is represented by angular limits appropriate to its rays.

Derivation of the Polar Jacobian

A small polar element has radial length approximately dr and arc length approximately r dθ. Therefore its area is approximately r dr dθ, giving dA=r dr dθ. This geometric argument also explains why area increases with distance from the origin.

Example 3: Area of a Circle

A = ∫₀^2π ∫₀^a r dr dθ
= ∫₀^2π (a²/2)dθ
= πa²

Example 4: Radially Varying Density

A circular plate of radius a has density ρ(r)=ρ₀ r/a, where ρ₀ is the density at the rim. Its mass is

m = ∫₀^2π∫₀^a (ρ₀ r/a)r dr dθ
= (2πρ₀/a)(a³/3)
= 2πρ₀a²/3

The units are consistent because ρ₀ has units kg/m² and a² has units m².

5. Triple Integrals in Cartesian Coordinates

For a solid V and continuous density or field f(x,y,z),

∭_V f(x,y,z)dV

where, in Cartesian coordinates,

dV = dx dy dz

The volume is ∭_V 1dV, and mass with volume density ρ(x,y,z) is

m = ∭_V ρ(x,y,z)dV

For a rectangular block, limits are constant. For a solid under z=g(x,y) and above a region R,

∭_V f dV = ∬_R ∫_z₁(x,y)^z₂(x,y) f(x,y,z) dz dA

For a three-dimensional body, changing integration order may substantially simplify the calculation.

6. Cylindrical Coordinates

Cylindrical coordinates extend polar coordinates into three dimensions:

x = r cosθ,
y = r sinθ,
z = z

The volume element is

dV = r dr dθ dz

This system is appropriate for cylinders, pipes, rotating components, and fields with axial symmetry. For a cylinder of radius a and height h,

0 ≤ r ≤ a, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h

Example 5: Mass of a Uniform Cylinder

For constant volume density ρ,

m = ∫₀^h∫₀^2π∫₀^a ρr dr dθ dz
= ρ(h)(2π)(a²/2)
= ρπa²h

This equals density multiplied by the familiar cylinder volume.

7. Spherical Coordinates

Spherical coordinates are defined by

x = ϱ sinφ cosθ,
y = ϱ sinφ sinθ,
z = ϱ cosφ

Here ϱ is distance from the origin, φ is measured downward from the positive z-axis, and θ is the azimuthal angle in the xy-plane. Their ranges are conventionally

ϱ ≥ 0, 0 ≤ φ ≤ π, 0 ≤ θ < 2π

The volume element is

dV = ϱ² sinφ dϱ dφ dθ

The factor ϱ² sinφ is the spherical Jacobian. Spherical coordinates are particularly effective for spheres, cones, and radially symmetric fields.

Example 6: Volume of a Sphere

For a sphere of radius a,

V = ∫₀^2π∫₀^π∫₀^a ϱ² sinφ dϱ dφ dθ

Separating factors gives

V = (2π)(2)(a³/3) = 4πa³/3

8. General Coordinate Transformations and Jacobians

Suppose

x = x(u,v), y = y(u,v)

The two-dimensional Jacobian is

∂(x,y)/∂(u,v) = | x_u x_v ; y_u y_v |
= x_u y_v − x_v y_u

The change-of-variables theorem states

∬_R f(x,y)dA = ∬_S f(x(u,v),y(u,v)) |∂(x,y)/∂(u,v)| du dv

The absolute value is required because area is nonnegative. The transformation should be continuously differentiable and one-to-one, except possibly on boundary sets of zero area. In higher dimensions, the Jacobian is the determinant of the derivative matrix:

dV = |∂(x,y,z)/∂(u,v,w)| du dv dw

For cylindrical and spherical coordinates, the determinants produce r and ϱ² sinφ, respectively.

9. Engineering and Probability Applications

Energy

If u(x,y) is energy per unit area, total energy stored in a surface is

E = ∬_R u(x,y)dA

For a three-dimensional energy density u(x,y,z) in J/m³,

E = ∭_V u(x,y,z)dV

Probability

A joint probability density function f_X,Y(x,y) must satisfy

f_X,Y(x,y) ≥ 0, ∬_R f_X,Y(x,y)dA = 1

The probability of an event represented by region A is

P((X,Y)∈A) = ∬_A f_X,Y(x,y)dA

Under a transformation, probability density must transform with the inverse Jacobian. If (U,V)=T(X,Y), then

f_U,V(u,v) = f_X,Y(x(u,v),y(u,v)) |∂(x,y)/∂(u,v)|

This preserves total probability and is valid where the transformation is suitably differentiable and one-to-one.

10. Applied Case: Inertia of a Robotic Turntable

Background

Consider an illustrative robotic turntable modelled as a uniform solid cylindrical disk of radius 0.30 m, thickness 0.04 m, and mass density 7800 kg/m³. The disk rotates about its central vertical axis.

Problem

Determine its mass and mass moment of inertia about the axis of rotation.

Analysis and Approach

Cylindrical coordinates are appropriate. The volume limits are 0≤r≤a, 0≤θ≤2π, and −h/2≤z≤h/2. The mass is

m = ∫_−h/2^h/2∫₀^2π∫₀^a ρ r dr dθ dz = ρπa²h

Substitution gives

m = 7800π(0.30)²(0.04) ≈ 88.0 kg

The distance from the rotation axis is r, so the mass moment of inertia is

I_z = ∭_V r²ρ dV
= ∫∫∫ ρr³ dr dθ dz
= (1/2)ρπa⁴h
= (1/2)ma²

Therefore,

I_z ≈ (1/2)(88.0)(0.30)² = 3.96 kg·m²

Outcome and Learning Points

The result can be used in motor sizing, acceleration prediction, and control-system simulation. The calculation demonstrates three principles: choose coordinates that match geometry, include the Jacobian r, and check units. Since kg·m² is the correct unit for rotational inertia, the result is dimensionally credible. Real designs must additionally account for hubs, fasteners, nonuniform material, machining tolerances, and structural flexibility.

11. Numerical and Practical Considerations

Analytical integration may be difficult when boundaries or densities are measured experimentally. Numerical quadrature, finite-element integration, and computer algebra can then be used. However, software does not determine correct physical limits automatically. Engineers should:

plot or inspect the domain;
verify coordinate ranges and singularities;
compare with a known special case;
check units and orders of magnitude;
refine numerical meshes until results stabilize;
document density, geometry, and boundary assumptions.

Coordinate singularities such as r=0 in polar coordinates or φ=0 in spherical coordinates are usually representation issues rather than physical singularities. A genuinely singular field requires separate convergence analysis. In robotics and automated engineering workflows, AI-assisted symbolic or numerical tools can suggest transformations, but results must be independently validated because an incorrect region description produces a precise-looking but invalid answer.

Chapter Summary

Double integrals accumulate fields over planar regions, while triple integrals accumulate them over solids. Cartesian coordinates are generally convenient for rectangular and linearly bounded regions. Polar coordinates use x=r cosθ, y=r sinθ and dA=r dr dθ; cylindrical coordinates use dV=r dr dθ dz; spherical coordinates use dV=ϱ² sinφ dϱ dφ dθ. Jacobians quantify area or volume scaling under transformations and must be included with absolute value in change-of-variables formulas. These tools support calculations of area, mass, centroids, moments of inertia, energy, and joint probabilities. Reliable engineering practice combines geometric reasoning, dimensional checks, suitable coordinate choice, and validation of analytical or numerical results.

Key Terms

Double integral: Integral of a scalar field over a two-dimensional region.

Triple integral: Integral of a scalar field over a three-dimensional solid.

Iterated integral: Multiple integral evaluated one variable at a time.

Region of integration: The area or volume over which integration occurs.

Fubini's theorem: Result permitting suitable multiple integrals to be evaluated as iterated integrals.

Polar coordinates: Planar coordinates using radius and angle.

Cylindrical coordinates: Three-dimensional coordinates extending polar coordinates along an axis.

Spherical coordinates: Three-dimensional coordinates using radial distance and two angles.

Jacobian: Determinant describing local scaling under a coordinate transformation.

Area element: Infinitesimal area, such as dA.

Volume element: Infinitesimal volume, such as dV.

Surface density: Mass per unit area.

Volume density: Mass per unit volume.

Centroid: Geometric or mass-weighted centre of a body.

Moment of inertia: Measure of resistance to angular acceleration.

Joint probability density: Density describing two or more random variables together.

Change-of-variables theorem: Rule for transforming an integral between coordinate systems.

Coordinate singularity: A point where a coordinate description degenerates.

Review Questions

Short Answer Questions

What physical quantity does ∬_R 1dA represent?
Why does the polar area element contain the factor r?
State the cylindrical-coordinate volume element.
Define the Jacobian of a two-variable transformation.
What units does a mass moment of inertia have?

Long Answer Questions

Explain how a non-rectangular planar region is converted into an iterated Cartesian integral.
Derive the polar-coordinate expression for the area of a disk.
Compare cylindrical and spherical coordinates, including their volume elements and typical engineering applications.
Derive the mass and centroid formulas for a variable-density lamina.
Explain how a Jacobian preserves the value of a probability integral under coordinate transformation.

Analytical / Application Questions

Evaluate the area inside the circle x²+y²≤4 using polar coordinates.
A solid occupies a cylinder of radius 0.2 m and height 0.5 m with density ρ(r)=1000(1+r) kg/m³, where r is measured in metres. Set up and evaluate its mass using cylindrical coordinates.
A robot arm component has a planar density that increases with distance from its pivot. Explain how you would calculate its mass and rotational inertia, including the coordinate system, Jacobian, limits, and required validation checks.

Multiple Choice Questions with Answers

Question 1

For polar coordinates, which area element is correct?

A. dA=dr dθ

B. dA=r dr dθ

C. dA=r² dr dθ

D. dA=sinθ dr dθ

Correct Answer: B. dA=r dr dθ

Question 2

The mass of a lamina with surface density ρ(x,y) is

A. ∬_R ρ dA

B. ∬_R ρ dx

C. ∫_R ρ ds

D. ∭_R ρ dV

Correct Answer: A. ∬_R ρ dA

Question 3

Which spherical-coordinate volume element is correct under the convention used in this chapter?

A. dV=ϱ dϱ dφ dθ

B. dV=ϱ² dϱ dφ dθ

C. dV=ϱ² sinφ dϱ dφ dθ

D. dV=sinφ dϱ dφ dθ

Correct Answer: C. dV=ϱ² sinφ dϱ dφ dθ

Question 4

The polar moment of inertia of a lamina about the origin is

A. ∬_R (x+y)ρ dA

B. ∬_R (x²+y²)ρ dA

C. ∬_R xyρ dA

D. ∬_R ρ dA

Correct Answer: B. ∬_R (x²+y²)ρ dA

Question 5

For a disk of radius a, the polar-coordinate radial limits are

A. −a≤r≤a

B. 0≤r≤a

C. 0≤r≤a²

D. 1≤r≤a

Correct Answer: B. 0≤r≤a

Question 6

The volume of a sphere of radius a obtained by spherical integration is

A. 2πa³/3

B. πa²

C. 4πa³/3

D. 4πa²

Correct Answer: C. 4πa³/3

Question 7

For the transformation x=x(u,v), y=y(u,v), the Jacobian is

A. The sum of x and y

B. The determinant of the partial-derivative matrix

C. The product uv only

D. The distance from the origin

Correct Answer: B. The determinant of the partial-derivative matrix

Question 8

A uniform cylinder has radius a and height h. Its volume is

A. 2πah

B. πa²h