Multiple Integrals

In one variable, $\int_a^b f\,dx$ is the area under the curve, built as a limit of sums of thin rectangles $f(x_i)\,\Delta x$. In two variables the idea moves up a dimension: the double integral $\iint_D f\,dA$ is the volume between the graph of $f(x,y)$ and the domain $D$ in the plane.

Construction (Riemann sums). Partition $D$ into many cells of area $\Delta A$, pick a point $(x_i,y_i)$ in each, form the sum of small volumes $f(x_i,y_i)\,\Delta A$, and let the cell size go to zero:

Useful special cases. With $f=1$, $\iint_D dA$ is the area of $D$. If $f\geq0$ it is the volume of the solid under the graph. If $f$ is a density, it is the mass. The practical challenge is not the definition but computing it: this is where Fubini comes in.

Fubini's theorem reduces a double integral to two single integrals "one inside the other". On a rectangle $R=[a,b]\times[c,d]$:

How to do it. In the inner integral you integrate with respect to one variable holding the other constant, obtaining a function of the remaining variable; then you integrate that. It is like slicing the solid into parallel slabs, computing the area of each slab, and summing the areas.

When the order can be swapped. If $f$ is continuous on the rectangle (or more generally if $\iint|f|<\infty$), both orders give the same result. But choosing the right order can make an integral easy instead of impossible: sometimes $\int e^{y^2}\,dy$ has no closed form, while swapping the order it does (see examples).

The domain is rarely a rectangle. For more general regions we use normal domains, where one variable lies between two functions of the other.

• $y$-normal domain: $D=\{a\leq x\leq b,\ g_1(x)\leq y\leq g_2(x)\}$ — for each fixed $x$, $y$ runs between two curves. Integrate in $y$ first, then $x$.
• $x$-normal domain: $x$ between two functions of $y$. Integrate in $x$ first, then $y$.

Swapping the order: the technique. Sometimes the inner integral is uncomputable in one order but immediate in the other. To swap: (1) draw the domain from the given bounds; (2) re-read it swapping the roles of the variables; (3) rewrite the new bounds. This turns $\int_0^1\!\int_x^1 e^{y^2}\,dy\,dx$ (impossible) into $\int_0^1\!\int_0^y e^{y^2}\,dx\,dy$ (trivial). Drawing the domain is almost always the key.

When the domain or the function has circular symmetry, Cartesian coordinates are awkward. We switch to $x=r\cos\theta$, $y=r\sin\theta$.

The area element changes. The little rectangle $dx\,dy$ becomes an "annular sector" of area $r\,dr\,d\theta$: the factor $r$ is the Jacobian determinant of the transformation. Intuitively, at a larger radius the same $d\theta$ spans a longer arc, so the area grows with $r$.

When to use them. Disks, rings, sectors; functions involving $x^2+y^2$ (which becomes $r^2$). A disk of radius $R$ has very simple bounds: $\theta\in[0,2\pi]$, $r\in[0,R]$. Forgetting the factor $r$ is the single most common mistake.

The triple integral $\iiint_V f\,dV$ extends the idea to three dimensions: it sums $f$ over a whole solid $V$. By Fubini it is computed by iterating three single integrals, slicing the solid along one direction:

"Slice by slice" strategy. Fix $(x,y)$ and integrate $z$ between the two surfaces bounding the solid (floor and ceiling); then integrate over the "shadow" of the solid in the $xy$-plane (a double integral). It is often best to project and reduce to a double integral.

What it computes. With $f=1$ it is the volume; with $f=\rho$ (density) the mass; with $f=\rho\,d^2$ the moment of inertia about an axis. The right coordinates (cylindrical, spherical) make many triple integrals computable.

In 3D there are two fundamental coordinate changes, each suited to a symmetry.

Cylindrical (polar in the plane + $z$): $x=r\cos\theta$, $y=r\sin\theta$, $z=z$. Jacobian $=r$, so $dV=r\,dr\,d\theta\,dz$. Ideal for cylinders, cones, paraboloids — anything with symmetry about an axis.

Spherical: $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, $z=\rho\cos\phi$, with $\rho\geq0$ (distance from the origin), $\phi\in[0,\pi]$ (angle from the north pole), $\theta\in[0,2\pi]$ (longitude). Jacobian $=\rho^2\sin\phi$:

Ideal for spheres, caps, and for functions depending on the distance from the origin (gravitational and electric potentials). The factor $\rho^2\sin\phi$ reflects that, far from the origin and near the equator, the same solid angle covers more volume.

Polar, cylindrical and spherical are special cases of one formula. For an invertible transformation $\Phi:(u,v)\mapsto(x,y)$:

Why $|\det J_\Phi|$ appears. An infinitesimal cell of area $du\,dv$ is deformed by $\Phi$ into a parallelogram of area $|\det J_\Phi|\,du\,dv$ (the Jacobian determinant is the area-scaling factor, §Vector functions). To preserve the integral's value, one must weight by this factor.

How to choose $\Phi$. Two aims: simplify the domain (make it a rectangle) or simplify the integrand. Example: for $\iint e^{(y-x)/(y+x)}$ over a triangle, the substitution $u=y-x$, $v=y+x$ straightens domain and function together. Always include the absolute value: $|\det J_\Phi|$, never signed.

Multiple integrals translate into formulas the physical quantities of an extended body with density $\rho$.

• Mass: density summed over the volume, $m=\iiint_V \rho\,dV$.
• Centroid (centre of mass): weighted average of the coordinates, $\bar x=\dfrac{1}{m}\iiint_V x\,\rho\,dV$ (and similarly for $\bar y,\bar z$). For homogeneous bodies it coincides with the geometric centroid.
• Moment of inertia about an axis: $I=\iiint_V d^2\,\rho\,dV$, with $d$ the distance from the axis. About the $z$-axis: $d^2=x^2+y^2$, very handy in cylindrical coordinates.

The moment of inertia measures the "resistance to being spun up": it is the rotational analogue of mass in Newton's second law ($\tau=I\alpha$). That is why choosing suitable coordinates (cylindrical for cylinders, spherical for spheres) is decisive: it turns nasty integrals into one-line computations.