Triple Integrals and Coordinate Systems

The double integral sums a density over a plane region; the triple integral sums a density over a solid. The definition, Fubini's Theorem, and the region descriptions all carry over with one more nesting. The new material is three coordinate systems for three-dimensional space and the general rule that ties them together: under a change of variables, the volume element scales by the absolute value of a determinant, the Jacobian. Cylindrical and spherical coordinates are the two most useful cases, and both are instances of that rule.

Surface area of a graph

Before the triple integral, one loose end from the double integral: the area of a curved surface $z = f (x, y)$ lying above a region $D$ . Cut $D$ into subrectangles; above each, the surface is nearly a slanted parallelogram, the piece of the tangent plane. Its two edge vectors are $(Δ x, 0, f_{x} Δ x)$ and $(0, Δ y, f_{y} Δ y)$ , and the area of the parallelogram they span is the magnitude of their cross product, $f_{x}^{2} + f_{y}^{2} + 1 Δ x Δ y$ . Summing and taking the limit gives the area.

The integrand exceeds $1$ wherever the surface tilts, so the surface is always at least as large as its shadow $D$ , with equality only for a flat horizontal piece. This formula is the two-variable analog of the arc-length integral $\int 1 + (f^{'})^{2} d x$ ; the parametric-surface form generalizes it to surfaces that are not graphs.

The triple integral

Let $f (x, y, z)$ be defined on a rectangular box $B = [a, b] \times [c, d] \times [r, s]$ . Partition each edge, cutting $B$ into sub-boxes $B_{ij k}$ of volume $Δ V = Δ x Δ y Δ z$ , choose a sample point in each, and form the triple Riemann sum. The limit defines the integral.

Fubini's Theorem again reduces the integral to nested single integrations, in any of the six orders:

B ∭ f (x, y, z) d V = a \int b c \int d r \int s f (x, y, z) d z d y d x .

Over a general bounded solid $E$ , enclose $E$ in a box, extend $f$ by zero, and the inner limits become surfaces. A solid $E$ is of type 1 if it sits between two graphs $z = u_{1} (x, y)$ and $z = u_{2} (x, y)$ over a plane region $D$ :

E ∭ f d V = D \iint u_{1} (x, y) \int u_{2} (x, y) f (x, y, z) d z d A .

The outer $D \iint$ is itself a double integral, evaluated as type I or type II. So the three limits are read off in order: the innermost from the two bounding surfaces, the middle two from the shadow $D$ of the solid on a coordinate plane.

A type 1 solid sits between a lower surface

z = u_{1}

and an upper surface

z = u_{2}

over its shadow

D

; the inner

z

-integral runs between the surfaces, the outer double integral sweeps

D

The most useful reading of $E ∭ 1 d V$ is the volume of $E$ , since the constant integrand $1$ counts $Δ V$ over every sub-box. With a density $ρ (x, y, z)$ , the same three definitions from the plane repeat: mass $m = E ∭ ρ d V$ , moments about the coordinate planes, and center of mass $(\overset{x}{ˉ}, \overset{y}{ˉ}, \overset{z}{ˉ})$ with $\overset{x}{ˉ} = \frac{1}{m} E ∭ x ρ d V$ , and similarly for the other two coordinates.

Cylindrical coordinates

Cylindrical coordinates keep $z$ and put the other two variables in polar form: $x = r cos θ$ , $y = r sin θ$ , $z = z$ . They fit solids with an axis of symmetry — cylinders, cones, paraboloids of revolution — because the axis becomes $r = 0$ and circular cross-sections become $r = const$ . The volume element inherits the polar factor: the base patch has area $r d r d θ$ and the box above it has height $d z$ , so

d V = r d z d r d θ .

The cylindrical volume element is a wedge of base

r d r d θ

and height

d z

; its volume

r d z d r d θ

carries the same factor

r

as the polar area element.

Spherical coordinates

Spherical coordinates measure a point by its distance from the origin and two angles: $ρ \geq 0$ the distance, $θ$ the same longitude angle as in cylindrical coordinates, and $ϕ \in [0, π]$ the angle down from the positive $z$ -axis (the colatitude). The conversions are

x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ,

and $ρ^{2} = x^{2} + y^{2} + z^{2}$ . Spheres are $ρ = const$ , cones from the origin are $ϕ = const$ , half-planes off the $z$ -axis are $θ = const$ . A spherical volume element is a box with edges $d ρ$ radially, $ρ d ϕ$ along a meridian, and $ρ sin ϕ d θ$ around a circle of latitude (radius $ρ sin ϕ$ ). Their product is the volume element.

The spherical volume element has radial edge

d ρ

, meridian edge

ρ d ϕ

, and latitude edge

ρ sin ϕ d θ

, giving

d V = ρ^{2} sin ϕ d ρ d ϕ d θ

The factor $ρ^{2} sin ϕ$ is the spherical volume element; $ρ^{2}$ because area grows with the square of the radius, $sin ϕ$ because circles of latitude shrink toward the poles.

The three factors separate because every limit is constant, recovering the familiar ball volume.

The general change of variables and the Jacobian

Polar, cylindrical, and spherical coordinates are three instances of one theorem. A change of variables is a map $T$ from a region $S$ in $uv$ -space to a region $D$ in $x y$ -space, $x = x (u, v)$ , $y = y (u, v)$ . A small rectangle in $uv$ -space with sides $Δ u$ and $Δ v$ maps to a small parallelogram in $x y$ -space spanned by the vectors $(x_{u}, y_{u}) Δ u$ and $(x_{v}, y_{v}) Δ v$ . The area of that parallelogram is the absolute value of a cross product, which is the determinant

\frac{\partial ( x , y )}{\partial ( u , v )} = det [x_{u} y_{u} x_{v} y_{v}] = x_{u} y_{v} - x_{v} y_{u} .

A transformation carries a unit cell in

uv

-space to a parallelogram in

x y

-space; the Jacobian determinant is the ratio of the two areas, the factor

d A

picks up under the change of variables.

Polar coordinates are the case $x = r cos θ$ , $y = r sin θ$ . The Jacobian is

\frac{\partial ( x , y )}{\partial ( r , θ )} = cos θ sin θ - r sin θ r cos θ = r cos^{2} θ + r sin^{2} θ = r,

matching the factor $r$ in $d A = r d r d θ$ . The three-variable version replaces the $2 \times 2$ determinant with a $3 \times 3$ one; computing it for the spherical map $x = ρ sin ϕ cos θ$ , etc., gives $\partial (x, y, z) / \partial (ρ, θ, ϕ) = ρ^{2} sin ϕ$ , and for the cylindrical map it gives $r$ . Each volume element is a Jacobian.

Coordinate map	Element	Jacobian	Best for
Cartesian	$d A = d x d y$	$1$	rectangles, straight boundaries
polar $(r, θ)$	$d A = r d r d θ$	$r$	disks, sectors, $x^{2} + y^{2}$ integrands
cylindrical $(r, θ, z)$	$d V = r d z d r d θ$	$r$	axial symmetry: cylinders, cones
spherical $(ρ, θ, ϕ)$	$d V = ρ^{2} sin ϕ d ρ d θ d ϕ$	$ρ^{2} sin ϕ$	radial symmetry: balls, spherical caps

The inner integral is $\int_{- v}^{v} e^{u / v} d u = v (e - e^{- 1})$ , and the outer one contributes $\frac{1}{2} \int_{1}^{2} v d v = \frac{3}{4}$ . Choosing $u$ and $v$ to straighten the boundary or simplify the integrand, then including the Jacobian, is the general method behind every named coordinate system.

Reading off the limits

Setting up a triple integral is entirely about describing the solid. The integrand rarely causes trouble; the limits do. The reliable procedure:

Sketch the solid and its shadow. Project $E$ onto a coordinate plane to get the double-integral region $D$ ; the two outer limits come from $D$ .
Innermost variable spans the two surfaces. For a type 1 solid the inner $z$ -limits are the lower and upper surfaces $u_{1} (x, y)$ and $u_{2} (x, y)$ .
Match the coordinate system to the symmetry. Axis of symmetry -> cylindrical; center of symmetry -> spherical; otherwise Cartesian or a custom Jacobian.
Never drop the Jacobian. Every non-Cartesian setup multiplies by $r$ , $ρ^{2} sin ϕ$ , or $∣ \partial (x, y, z) / \partial (u, v, w) ∣$ .¹

Stewart, §15.5 — Surface Area; §15.6 — Triple Integrals; §15.7 — Cylindrical Coordinates; §15.8 — Spherical Coordinates; §15.9 — Change of Variables in Multiple Integrals. The volume elements $r d z d r d θ$ and $ρ^{2} sin ϕ d ρ d θ d ϕ$ and their derivation as Jacobian determinants. ↩

Surface area of a graph

The triple integral

Cylindrical coordinates

Spherical coordinates

The general change of variables and the Jacobian

Reading off the limits

Footnotes