Multiple integrals

In short

A multiple integral extends integration from one variable to two or three, summing a quantity over a plane region or a solid instead of along a line. Double integrals give area, mass, centroids and moments of plane laminae; triple integrals give volume, mass and moments of inertia of solids; a change of variables (Jacobian) converts them into polar, cylindrical or spherical form where the region becomes simple. In Indian university B.Sc/B.E. syllabi (UGC CBCS Calculus and Multivariable Calculus papers, AICTE Mathematics-II) this is examined every year through change of order of integration, Jacobians and volume problems, and GATE/JAM/CSIR-NET ask one or two marks-heavy questions on it. It also underlies the integrals used in probability (joint densities), finite-element and CFD work at Indian engineering and analytics employers.

Explained

Double integral: definition and meaning

• Partition a closed bounded region R in the xy-plane into n sub-regions of areas ΔA₁…ΔAₙ, pick a point (xᵢ,yᵢ) in each, and form the Riemann sum Σ f(xᵢ,yᵢ)ΔAᵢ.
• If the sum tends to a unique limit as n → ∞ and each sub-region shrinks to a point (max diameter → 0), that limit is the double integral ∬_R f(x,y) dA.
• Existence: f continuous on a closed bounded R (or bounded with discontinuities only on a set of zero area) is sufficient.
• Geometric reading: for f ≥ 0, ∬_R f dA is the volume of the cylinder on base R under the surface z = f(x,y); with f = 1 it is the area of R.
• Properties: linearity, additivity over non-overlapping sub-regions, monotonicity (f ≤ g ⇒ ∬f ≤ ∬g), and |∬f| ≤ ∬|f|.
• Mean value form: ∬_R f dA = f(x₀,y₀) × Area(R) for some interior point, if f is continuous.

Evaluation as iterated integrals and Fubini's theorem

• Fubini's theorem: if f is continuous on R, the double integral equals either iterated integral, so the order of integration may be interchanged.
• Vertical strips (y inner): R bounded by y = g₁(x) below, y = g₂(x) above, between x = a and x = b → ∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f dy dx. Integrate in y treating x as constant.
• Horizontal strips (x inner): R bounded by x = h₁(y), x = h₂(y) between y = c and y = d → ∫_c^d ∫_{h₁(y)}^{h₂(y)} f dx dy.
• Rule for limits: inner limits may be functions of the outer variable; outer limits must be constants. A variable left in the final answer means the limits were set up wrongly.
• Over a rectangle [a,b]×[c,d] with f(x,y) = p(x)q(y), the integral factorises: (∫ₐᵇ p dx)(∫_c^d q dy).
• Change of order of integration (a standard 5–8 mark question): sketch R from the given limits, re-describe it with strips in the other direction, and rewrite the limits. Used when the inner integral is not elementary (e.g. ∫ e^{-y²} dy, ∫ (sin y)/y dy) or when the region needs splitting.
• If R is not of either simple type, split it into sub-regions, each simple, and add.

Change of variables and the Jacobian

• For x = x(u,v), y = y(u,v) mapping region R' in the uv-plane one-to-one onto R: dA = |J| du dv where J = ∂(x,y)/∂(u,v) is the determinant of the matrix of partials.
• ∬_R f(x,y) dx dy = ∬_{R'} f(x(u,v),y(u,v)) |J| du dv. |J| is the local area-magnification factor.
• Inverse rule: ∂(x,y)/∂(u,v) × ∂(u,v)/∂(x,y) = 1, so compute whichever determinant is easier.
• Polar: x = r cos θ, y = r sin θ, J = r, so dA = r dr dθ. Use whenever x² + y² appears, or the region is a circle, annulus, cardioid or sector.
• Polar limits: for a region swept by radii, r from the inner curve r = f₁(θ) to the outer r = f₂(θ), θ over the angular span.
• Triple integral Jacobians: cylindrical (r, θ, z): dV = r dr dθ dz; spherical (r, θ, φ) with x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ: dV = r² sinθ dr dθ dφ (θ = polar angle from the z-axis, φ = azimuth; some Indian texts swap the letters, so always state the convention used).
• Other common substitutions: u = x + y, v = x − y for parallelogram regions; x = au, y = bv to turn an ellipse into a unit circle (J = ab).

Triple integrals

• ∭_V f(x,y,z) dV is defined by the same limiting process over sub-volumes; with f = 1 it gives the volume of V.
• Evaluation as a triple iterated integral ∫∫∫ f dz dy dx: innermost limits from the lower surface z = ψ₁(x,y) to the upper z = ψ₂(x,y); the remaining double integral is over the projection of V on the xy-plane.
• Order can be chosen freely (6 possible orders) subject to Fubini; pick the order that keeps limits simple.
• Volume between two surfaces as a double integral: V = ∬_R [z_top − z_bottom] dA — often shorter than a triple integral.
• Volume of revolution also follows: rotating a plane region, or directly V = ∬ r dr dθ-type set-ups in cylindrical coordinates.

Applications: area, mass, centroid, moments

• Area of a plane region: A = ∬_R dA (Cartesian) or ∬ r dr dθ (polar).
• Mass of a lamina of surface density ρ(x,y) (kg/m²): m = ∬_R ρ dA; for a solid of density ρ(x,y,z) (kg/m³): m = ∭_V ρ dV.
• First moments and centroid: x̄ = (1/m)∬ xρ dA, ȳ = (1/m)∬ yρ dA. For uniform density ρ cancels and the centroid is purely geometric — this is how section centroids in IS 800 steel design and IS 456 RC design are derived.
• Moments of inertia (second moments): I_x = ∬ y²ρ dA, I_y = ∬ x²ρ dA, polar moment I_0 = I_x + I_y = ∬ (x²+y²)ρ dA. Units: kg·m² for mass moment, m⁴ for the geometric second moment of area used in beam bending.
• Radius of gyration: k = √(I/m).
• Average value of f over R: f̄ = (1/A)∬_R f dA — used for mean rainfall over a catchment, mean pressure over a plate.
• Surface area of z = f(x,y) over R: S = ∬_R √(1 + f_x² + f_y²) dA.
• Probability: for a joint density f(x,y), P((X,Y) ∈ R) = ∬_R f dA, and ∬_{ℝ²} f dA = 1.

Improper multiple integrals and standard results

• Over unbounded regions or with unbounded f, define as a limit over expanding bounded regions; convergence must be checked.
• Classic polar trick: (∫_0^∞ e^{-x²} dx)² = ∬_{first quadrant} e^{-(x²+y²)} dA = ∫_0^{π/2}∫_0^∞ e^{-r²} r dr dθ = π/4, giving ∫_0^∞ e^{-x²} dx = √π/2.
• Dirichlet's integral (frequently examined): over x, y, z ≥ 0 with x + y + z ≤ 1, ∭ x^{l−1} y^{m−1} z^{n−1} dV = Γ(l)Γ(m)Γ(n)/Γ(l+m+n+1).
• Beta–Gamma link: B(m,n) = ∫_0¹ x^{m−1}(1−x)^{n−1} dx = Γ(m)Γ(n)/Γ(m+n); Γ(n+1) = nΓ(n), Γ(1/2) = √π.
• Leibniz rule for differentiating under the integral sign is the companion tool when a parameter appears in the integrand.
• Green's theorem converts a line integral round a closed curve C into a double integral: ∮_C (M dx + N dy) = ∬_R (∂N/∂x − ∂M/∂y) dA; Gauss divergence theorem converts a closed surface integral into a triple integral ∭ (∇·F) dV. These are the bridge to vector calculus questions.
• Limitations: Fubini can fail for non-absolutely-convergent improper integrals; a non-one-to-one transformation or a vanishing Jacobian invalidates the change of variables; numerical quadrature (Simpson's rule applied twice, Gauss quadrature, Monte Carlo for high dimensions) is used when no closed form exists.

Key points

• Double integral — limit of Σ f ΔA over a plane region; equals volume under a surface when f ≥ 0.
• Iterated integral — evaluated inside-out; inner limits may contain the outer variable, outer limits must be constants.
• Fubini's theorem — for continuous f on a bounded region, dx dy and dy dx give the same value.
• Change of order — always sketch the region first; the answer's value never changes, only the ease of evaluation.
• f = 1 — the integral degenerates to area (double) or volume (triple).
• Jacobian J — determinant of partial derivatives; dA = |J| du dv is the area scale factor.
• Polar — dA = r dr dθ; the extra r is the most commonly forgotten factor in exams.
• Cylindrical — dV = r dr dθ dz, best for cylinders, cones and paraboloids about an axis.
• Spherical — dV = r² sinθ dr dθ dφ, best for spheres, cones with apex at origin, and x²+y²+z² integrands.
• Projection rule — in a triple integral, integrate z between the two bounding surfaces, then over the shadow of the solid.
• Mass — m = ∬ ρ dA or ∭ ρ dV; centroid = first moment ÷ mass.
• Moment of inertia — I_x = ∬ y²ρ dA; polar moment I_0 = I_x + I_y (perpendicular axis idea).
• Surface area — S = ∬ √(1+f_x²+f_y²) dA for an explicit surface z = f(x,y).
• Dirichlet integral — gamma-function formula for integrals over the simplex x+y+z ≤ 1.
• Gaussian integral — ∫_0^∞ e^{-x²} dx = √π/2, proved by switching to polar in two dimensions.
• Green's and Gauss's theorems — reduce line integrals to double integrals and surface integrals to triple integrals.
• Symmetry — an odd integrand over a symmetric region vanishes; use it to halve or quarter the work.
• Numerical fallback — repeated Simpson/Gauss quadrature or Monte Carlo when the integral has no closed form.

Formulas and examples

Example 1 — straight iterated integral. Evaluate I = ∫_0¹ ∫_0² (x² + y²) dy dx.

• Inner: ∫_0² (x² + y²) dy = [x²y + y³/3]_0² = 2x² + 8/3.
• Outer: ∫_0¹ (2x² + 8/3) dx = 2/3 + 8/3 = 10/3 ≈ 3.333.

Example 2 — change of order of integration. Evaluate I = ∫_0¹ ∫_x¹ (sin y)/y dy dx (inner integral has no elementary antiderivative).

• Region: 0 ≤ x ≤ 1, x ≤ y ≤ 1 — the triangle with vertices (0,0), (0,1), (1,1), i.e. y from 0 to 1 and x from 0 to y.
• Reversed: I = ∫_0¹ ∫_0^y (sin y)/y dx dy = ∫_0¹ (sin y)/y × y dy = ∫_0¹ sin y dy.
• I = [−cos y]_0¹ = 1 − cos 1 = 1 − 0.540302 = 0.4597.

Example 3 — polar coordinates and a quarter circle. Find ∬_R √(x² + y²) dA over the quarter disc x² + y² ≤ 9, x ≥ 0, y ≥ 0.

• Switch to polar: integrand = r, dA = r dr dθ, so I = ∫_0^{π/2} ∫_0³ r·r dr dθ.
• ∫_0³ r² dr = 27/3 = 9; ∫_0^{π/2} dθ = π/2.
• I = 9 × π/2 = 4.5π ≈ 14.137 (units: m³ if lengths are in metres).

Example 4 — volume of a solid by triple integration. Volume cut from the paraboloid z = x² + y² below the plane z = 4.

• Cylindrical: z runs from r² to 4, r from 0 to 2 (where r² = 4), θ from 0 to 2π.
• V = ∫_0^{2π}∫_0² (4 − r²) r dr dθ = 2π ∫_0² (4r − r³) dr = 2π [2r² − r⁴/4]_0² = 2π (8 − 4) = 8π ≈ 25.13 m³.
• Check: half the volume of the circumscribing cylinder πr²h = π(4)(4) = 16π — correct, a paraboloid encloses half its cylinder.

Example 5 — mass, centroid and moment of inertia of a lamina. A triangular steel gusset plate occupies 0 ≤ x ≤ 2 m, 0 ≤ y ≤ (2 − x) m, with areal density ρ = 40 kg/m² (about a 5 mm mild-steel plate at 7,850 kg/m³).

• Area = ½ × 2 × 2 = 2 m² → m = 40 × 2 = 80 kg.
• ∬ x dA = ∫_0² ∫_0^{2−x} x dy dx = ∫_0² x(2−x) dx = [x² − x³/3]_0² = 4 − 8/3 = 4/3.
• x̄ = (ρ × 4/3)/m = (40 × 4/3)/80 = 0.667 m; by symmetry of the region ȳ = 0.667 m (matches the centroid of a right triangle at one-third of each leg).
• I_x = ρ∬ y² dA = 40 ∫_0² ∫_0^{2−x} y² dy dx = 40 ∫_0² (2−x)³/3 dx = (40/3)[−(2−x)⁴/4]_0² = (40/3)(4) = 53.33 kg·m²; similarly I_y = 53.33 kg·m², so I_0 = 106.7 kg·m².
• Radius of gyration k = √(53.33/80) = 0.816 m.

Where it is used

• Structural and mechanical design: section properties (centroid, I, section modulus) used in IS 800:2007 steel design and IS 456:2000 RC design; stress integration over a cross-section in Indian design offices (L&T, Tata Consulting Engineers).
• CFD, FEA and simulation roles: element stiffness and mass matrices are area/volume integrals evaluated by Gauss quadrature; used at Ansys, Altair, Siemens, Tata Technologies and ISRO/DRDO labs.
• Water resources and environment: mean rainfall over a catchment, reservoir volume from contour areas, pollutant load over an area — CWC and state irrigation departments; the IS code on rainfall data analysis supports the Thiessen/isohyetal area-weighting that these integrals formalise.
• Data science, actuarial and quant roles: joint probability densities, expected values E[g(X,Y)] = ∬ g f dA, bivariate normal probabilities — asked in IRDAI-track actuarial exams and analytics interviews at Indian banks and insurers.
• Electrical and electronics: charge, flux and field computations via Gauss's divergence theorem; power dissipation integrated over a device cross-section.
• Image processing and geospatial work: integrating pixel intensity or land-use value over a region, area computation from satellite imagery at ISRO/NRSC and survey firms.

Interview questions

• Q: What does a double integral physically represent? A: The limit of a sum of f(x,y)ΔA over a region; for f ≥ 0 it is the volume under the surface z = f(x,y), and for f = 1 it is the area of the region.
• Q: When can you interchange the order of integration? A: When the integrand is continuous on a closed bounded region (Fubini's theorem), or more generally when the double integral of |f| is finite; the limits must be rewritten to describe the same region.
• Q: Why does dA become r dr dθ in polar coordinates? A: The Jacobian of x = r cosθ, y = r sinθ is r, so a small polar element has area r dr dθ — physically, arc length r dθ times radial width dr.
• Q: What goes wrong if the Jacobian is zero somewhere? A: The transformation is not locally one-to-one there, so the change-of-variables formula may fail; the point (e.g. r = 0 in polar) must be handled as a limiting or measure-zero case.
• Q: How would you find the volume between two surfaces? A: Project the solid onto a coordinate plane to get region R and integrate the difference of the upper and lower surfaces: V = ∬_R (z_top − z_bottom) dA.
• Q: Why would you switch to spherical coordinates? A: When the region is a sphere, hemisphere or cone about the origin, or the integrand involves x²+y²+z² — the limits become constants and dV = r² sinθ dr dθ dφ.
• Q: How is ∫_0^∞ e^{-x²} dx evaluated? A: Square it to get a double integral over the first quadrant, switch to polar to get ∫_0^{π/2}∫_0^∞ e^{-r²} r dr dθ = π/4, so the integral is √π/2.
• Q: What is the difference between the mass moment of inertia and the second moment of area? A: The mass moment includes density (kg·m²) and governs rotational dynamics; the second moment of area is purely geometric (m⁴) and governs bending stiffness.
• Q: State Green's theorem and one use. A: ∮_C (M dx + N dy) = ∬_R (∂N/∂x − ∂M/∂y) dA; with M = −y/2, N = x/2 it gives the area enclosed by a closed curve, the basis of the planimeter and of polygon-area formulas in survey software.
• Q: Can the order of integration change the value of an integral? A: For absolutely convergent integrals, no; for conditionally convergent improper double integrals the two iterated values can differ, which is why Fubini requires integrability of |f|.
• Q: How do you handle a region that is neither vertically nor horizontally simple? A: Split it into non-overlapping sub-regions that are each simple, integrate separately and add, using additivity of the double integral.
• Q: If a closed-form answer does not exist, what do you do? A: Use numerical quadrature — repeated Simpson's or Gauss–Legendre rules on a mapped square, or Monte Carlo sampling when the dimension is high.

Common mistakes

• Omitting the Jacobian factor — writing ∬ f dr dθ instead of ∬ f r dr dθ, or dropping r² sinθ in spherical coordinates.
• Keeping a variable in the outer limits, so the final answer still contains x or y.
• Changing the order of integration without sketching the region, which almost always produces wrong limits.
• Not splitting a region when the bounding curves change (for example where a line and a parabola intersect), so one set of limits is used for the whole region.
• Confusing the polar angle and azimuth in spherical coordinates, or using degrees instead of radians in the θ-limits.
• Mixing units — density in g/cm³ with lengths in metres — when computing mass, centroid or moment of inertia.
• Assuming symmetry cancels an integral without checking that both the region and the integrand have that symmetry.

Related topics

Integral calculus · Applications of derivatives · Vector calculus: Green's, Stokes' and Gauss's theorems · Beta and gamma functions · Partial differentiation and Jacobians · Numerical integration