Multiple Integrals

1. Introduction

A single integral $\int_a^b f(x)\,dx$ accumulates a quantity along an interval — area under a curve, total distance, total mass of a thin rod. But many quantities live over two- or three-dimensional regions: the volume beneath a surface, the mass of a flat plate of varying density, the average temperature in a solid body, the center of mass of an oddly shaped object. To accumulate over regions of higher dimension we need multiple integrals.

The double integral $\iint_R f(x,y)\,dA$ sums the values of $f$ over a planar region $R$, and the triple integral $\iiint_E f(x,y,z)\,dV$ sums over a solid region $E$. The conceptual leap from single integrals is small — we still chop the domain into tiny pieces, multiply by the integrand, and add — but the execution requires two new skills: describing two- and three-dimensional regions with correct limits of integration, and choosing a coordinate system (rectangular, polar, cylindrical, or spherical) that matches the geometry.

This article builds the theory from Riemann sums and Fubini's theorem through general (Type I/II) regions, change of order of integration, polar coordinates and the Jacobian, triple integrals, cylindrical and spherical coordinates, and the major applications: area, volume, mass, center of mass, moments of inertia, and average value. Mastery means you can set up and evaluate any reasonable multiple integral and, crucially, recognize when switching coordinates or order transforms an impossible integral into a routine one.

Multiple integrals are the workhorse of applied mathematics: computing probabilities over regions, finding centers of mass in engineering, evaluating electrostatic potentials, and determining volumes of irregular solids. The Jacobian factor is not a mere bookkeeping device — it encodes the geometric fact that coordinate grids are distorted by nonlinear transformations, and the integral must weight each patch by its true area or volume.

2. Core Concepts

2.1 The Double Integral as a Limit of Riemann Sums

Partition a rectangle $R=[a,b]\times[c,d]$ into a grid of subrectangles, each of area $\Delta A$, and pick a sample point $(x_{ij}^*,y_{ij}^*)$ in each. The double Riemann sum is $\sum_i\sum_j f(x_{ij}^*,y_{ij}^*)\,\Delta A$. The double integral is the limit as the grid is refined: $\iint_R f(x,y)\,dA = \lim_{\|P\|\to 0}\sum_{i}\sum_{j} f(x_{ij}^*,y_{ij}^*)\,\Delta A.$ If $f$ is continuous on $R$ this limit exists. When $f(x,y)\ge 0$ the integral equals the volume of the solid lying above $R$ and below the surface $z=f(x,y)$. With $f\equiv 1$ it returns the area of $R$.

2.2 Fubini's Theorem and Iterated Integrals

We never evaluate the limit directly. Fubini's theorem reduces a double integral to two ordinary integrals performed in succession (an iterated integral). Over a rectangle, $\iint_R f(x,y)\,dA = \int_a^b\!\!\int_c^d f(x,y)\,dy\,dx = \int_c^d\!\!\int_a^b f(x,y)\,dx\,dy.$

Proof sketch (continuous $f$ on a rectangle). Slice the solid under $z=f(x,y)$ with planes $x=\text{const}$. Each slice has cross-sectional area $A(x)=\int_c^d f(x,y)\,dy$. The volume is $\int_a^b A(x)\,dx = \int_a^b\int_c^d f(x,y)\,dy\,dx$. Symmetry of the slicing argument gives the reverse order.

2.3 General Regions: Type I and Type II

Most regions are not rectangles. We describe a region in one of two ways:

• Type I (vertically simple): bounded below and above by curves of $x$, $R = \{(x,y) : a\le x\le b,\;\; g_1(x)\le y\le g_2(x)\}, \qquad \iint_R f\,dA = \int_a^b\!\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx.$
• Type II (horizontally simple): bounded left and right by curves of $y$, $R = \{(x,y) : c\le y\le d,\;\; h_1(y)\le x\le h_2(y)\}, \qquad \iint_R f\,dA = \int_c^d\!\!\int_{h_1(y)}^{h_2(y)} f\,dx\,dy.$

The inner limits may depend on the outer variable, but the outer limits must be constants. Sketching the region is essential to read off correct limits.

2.4 Changing the Order of Integration

A given region can usually be described both ways, and the resulting iterated integrals are equal. Switching the order is a key technique: an integral like $\int_0^1\!\int_x^1 e^{y^2}\,dy\,dx$ is impossible as written (no elementary antiderivative of $e^{y^2}$ in $y$), but reversing to $\int_0^1\!\int_0^y e^{y^2}\,dx\,dy$ becomes elementary. Always redraw the region when changing order — never just swap the limits mechanically.

2.5 Polar Coordinates and the Area Element

For regions with circular symmetry (disks, annuli, sectors) or integrands involving $x^2+y^2$, polar coordinates $x=r\cos\theta$, $y=r\sin\theta$ are transformative. The critical fact is the area element: $dA = r\,dr\,d\theta.$ The extra factor $r$ is the Jacobian of the transformation — it accounts for the fact that polar "rectangles" $[r,r+dr]\times[\theta,\theta+d\theta]$ have area $\approx r\,dr\,d\theta$, growing with distance from the origin.

2.6 The General Change of Variables and the Jacobian

For a transformation $x=x(u,v)$, $y=y(u,v)$, the area element transforms by the absolute Jacobian determinant: $\frac{\partial(x,y)}{\partial(u,v)} = \det\begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\[4pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v}\end{bmatrix}, \qquad dA = \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv.$

Why the Jacobian appears. A small rectangle $du\times dv$ in the $(u,v)$-plane maps to a parallelogram in the $(x,y)$-plane spanned by $\langle\partial x/\partial u,\partial y/\partial u\rangle\,du$ and $\langle\partial x/\partial v,\partial y/\partial v\rangle\,dv$. Its area is $|\partial(x,y)/\partial(u,v)|\,du\,dv$.

2.7 Triple Integrals and 3D Coordinate Systems

A triple integral $\iiint_E f\,dV$ extends the idea to a solid $E$, evaluated as an iterated integral over (typically) $z$ then $y$ then $x$, with appropriate variable limits. Two specialized systems streamline 3D problems:

• Cylindrical coordinates $(r,\theta,z)$ with $x=r\cos\theta$, $y=r\sin\theta$, $z=z$ and volume element $dV = r\,dz\,dr\,d\theta.$ Ideal for solids with an axis of symmetry (cylinders, cones, paraboloids).
• Spherical coordinates $(\rho,\phi,\theta)$ where $\rho\ge 0$ is the distance to the origin, $\phi\in[0,\pi]$ is the angle from the positive $z$-axis, and $\theta$ is the usual azimuthal angle. Then $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, $z=\rho\cos\phi$, and $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta.$ Ideal for balls, spherical shells, and integrands involving $x^2+y^2+z^2$.

2.8 Physical Applications

• Mass: $m = \iint_R \delta(x,y)\,dA$ (or $\iiint_E \delta\,dV$) for density $\delta$.
• Center of mass: $\bar x = \frac{1}{m}\iint_R x\,\delta\,dA$, $\bar y = \frac{1}{m}\iint_R y\,\delta\,dA$.
• Moments of inertia: $I_x = \iint_R y^2\delta\,dA$ (about the $x$-axis); analogous formulas in 3D.
• Average value: $\bar f = \frac{1}{A(R)}\iint_R f\,dA$.