Triple Integrals and Change of Variables

Learning Objectives

  • Interpret a triple integral as scalar accumulation over an ordinary three-dimensional solid.
  • Construct rectangular-coordinate bounds from projections and bounding surfaces.
  • Convert between rectangular, cylindrical, and spherical coordinates and use the correct volume Jacobians.
  • Compute volume, mass, center of mass, and moments of inertia of solids.
  • Select coordinates that simplify a solid's geometry and integrand.
  • State and apply the change-of-variables theorem with the absolute Jacobian determinant.
  • Compute two- and three-dimensional Jacobians for general transformations.
  • Transform non-polar regions, bounds, integrands, and differential elements correctly.

A triple integral accumulates a scalar quantity throughout a three-dimensional region EE. It does not require a four-dimensional geometric interpretation: ∭E1 dV\iiint_E 1\,dV is the ordinary three-dimensional volume of EE, while ∭Ef dV\iiint_E f\,dV accumulates the scalar field ff over that volume.

Triple Integrals over Solids

Triple Integral

For an integrable scalar field f(x,y,z)f(x,y,z) over a solid E⊂R3E\subset\mathbb{R}^3, the triple integral ∭Ef dV\iiint_E f\,dV is the limit of sums of the form ∑f(xi∗,yi∗,zi∗) ΔV\sum f(x_i^*,y_i^*,z_i^*)\,\Delta V.

Triple Integral

Scalar accumulation over a three-dimensional solid.

∭Ef(x,y,z) dV.\iiint_E f(x,y,z)\,dV.

Geometric and physical meanings

  • If f=1f=1, then ∭E1 dV\iiint_E 1\,dV is the 3D volume of EE.
  • If f=ρ(x,y,z)f=\rho(x,y,z) is volumetric mass density, then the integral is mass.
  • If ff is a signed field, positive and negative contributions are accumulated with sign.

Rectangular Coordinates and Bounds Construction

For the order dz dy dxdz\,dy\,dx, first project the solid onto the xyxy-plane. Describe that projection by a≤x≤ba\le x\le b and g1(x)≤y≤g2(x)g_1(x)\le y\le g_2(x). Then, at each point (x,y)(x,y) in the projection, describe the lower and upper surfaces in the zz-direction.

Typical dz dy dx Setup

A solid bounded vertically by two surfaces.

∭Ef dV=∫ab∫g1(x)g2(x)∫u1(x,y)u2(x,y)f(x,y,z) dz dy dx.\iiint_E f\,dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z)\,dz\,dy\,dx.

Constructing Triple-Integral Bounds

  • Choose an integration order.
  • Project the solid onto the coordinate plane associated with the two outer variables.
  • Describe the projected region using valid two-dimensional bounds.
  • For each point in that projection, identify the lower and upper surfaces for the inner variable.
  • Split the solid if one set of bounds cannot describe it without overlap or omission.
  • Perform a dimensional and geometric check before integrating.
Project the solid before setting nested bounds

For the illustrated solid, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 − x, and 0 ≤ z ≤ 2 − x/2 − y/2. The dashed footprint is the xy projection and a vertical slice identifies the inner z limits. The schematic is not to scale.

A solid over a triangular xy projection has z bounds from zero to a sloping upper plane; the projection gives the outer bounds for dz dy dx.

Cylindrical Coordinates

Cylindrical coordinates retain the vertical coordinate zz while using polar coordinates in the xyxy-plane. They are especially effective for cylinders, cones, paraboloids of revolution, and solids with circular projections.

Cylindrical Coordinate Relations

  • x=rcos⁡θx=r\cos\theta
  • y=rsin⁡θy=r\sin\theta
  • z=zz=z
  • x2+y2=r2x^2+y^2=r^2
  • r≥0r\ge0

Cylindrical Volume Element

Jacobian-scaled volume differential.

dV=r dz dr dθdV=r\,dz\,dr\,d\theta

for the displayed order. Any consistent permutation of dzdz, drdr, and dθd\theta is possible when the corresponding bounds are rewritten correctly.

The factor r is mandatory

The cylindrical transformation inherits the polar area scaling in the xyxy-plane, so omitting rr changes the value and the units of the integral.

Why cylindrical volume includes the factor r

At radius r, the short tangential side is approximately r dθ. Multiplying it by dr and dz gives dV = r dr dθ dz; the order of integration may change only when the bounds are rewritten consistently.

An annular sector with radial thickness dr, angular arc approximately r d theta, and vertical height dz forms a local cylindrical volume element.

Spherical Coordinates

This course uses ρ\rho for distance from the origin, θ\theta for the azimuth angle in the xyxy-plane, and ϕ\phi for the polar angle measured from the positive zz-axis.

Spherical Coordinate Relations

  • x=ρsin⁡ϕcos⁡θx=\rho\sin\phi\cos\theta
  • y=ρsin⁡ϕsin⁡θy=\rho\sin\phi\sin\theta
  • z=ρcos⁡ϕz=\rho\cos\phi
  • x2+y2+z2=ρ2x^2+y^2+z^2=\rho^2
  • r=ρsin⁡ϕr=\rho\sin\phi
  • ρ≥0\rho\ge0, usually 0≤θ≤2π0\le\theta\le2\pi, 0≤ϕ≤π0\le\phi\le\pi

Spherical Volume Element

Jacobian-scaled volume differential.

dV=ρ2sin⁡ϕ dρ dϕ dθdV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta

Angle convention

Always verify the convention used by a textbook or software package. Here ϕ\phi is measured from the positive zz-axis; under a different convention the conversion formulas and Jacobian expression may be written differently.

The three side lengths behind the spherical Jacobian

The local side lengths are approximately dρ radially, ρ dφ in the polar direction, and ρ sinφ dθ around the z-axis. Their product yields dV = ρ² sinφ dρ dφ dθ for the lesson's angle convention.

Spherical coordinates measure rho from the origin and phi from the positive z-axis; a local differential cell has side lengths d rho, rho d phi, and rho sin phi d theta.

Coordinate-Selection Strategy

Coordinate-selection guide

  • Use rectangular coordinates when bounding planes and coordinate-aligned boxes dominate.
  • Use cylindrical coordinates when x2+y2x^2+y^2, circular projections, cones about the zz-axis, or surfaces z=g(r)z=g(r) dominate.
  • Use spherical coordinates when spheres centered at the origin, cones ϕ=constant\phi=\text{constant}, or radial expressions x2+y2+z2x^2+y^2+z^2 dominate.
  • Prefer the system that simplifies both the region and the integrand; a symmetric region alone does not guarantee the simplest integral.
  • Check that the angular and radial bounds cover the solid exactly once.

Mass, Center of Mass, and Moments of Inertia

Let δ(x,y,z)≥0\delta(x,y,z)\ge0 denote volumetric mass density.

Mass of a Solid

Integral of volumetric density.

m=∭Eδ(x,y,z) dV.m=\iiint_E\delta(x,y,z)\,dV.

First Moments

Moments about the coordinate planes.

Myz=∭Exδ dV,Mxz=∭Eyδ dV,Mxy=∭Ezδ dV.M_{yz}=\iiint_E x\delta\,dV,\qquad M_{xz}=\iiint_E y\delta\,dV,\qquad M_{xy}=\iiint_E z\delta\,dV.

Center of Mass

Density-weighted average position.

xˉ=Myzm,yˉ=Mxzm,zˉ=Mxym.\bar{x}=\frac{M_{yz}}{m},\qquad \bar{y}=\frac{M_{xz}}{m},\qquad \bar{z}=\frac{M_{xy}}{m}.

Moments of Inertia of a Solid

Integrate squared perpendicular distance to each axis.

Ix=∭E(y2+z2)δ dV,I_x=\iiint_E(y^2+z^2)\delta\,dV,Iy=∭E(x2+z2)δ dV,I_y=\iiint_E(x^2+z^2)\delta\,dV,Iz=∭E(x2+y2)δ dV.I_z=\iiint_E(x^2+y^2)\delta\,dV.

The Change-of-Variables Theorem

A coordinate transformation changes both the description of the region and the local size of area or volume elements. The Jacobian determinant measures this local scaling. The absolute value is used for ordinary area and volume integrals because area and volume are nonnegative measures even if the transformation reverses orientation.

Two-Dimensional Theorem

2D Change of Variables

Let T(u,v)=(x(u,v),y(u,v))T(u,v)=(x(u,v),y(u,v)) be a continuously differentiable one-to-one transformation on a region SS (up to standard boundary exceptions), with nonzero Jacobian in the interior. If R=T(S)R=T(S), then

2D Change-of-Variables Theorem

Transform a double integral using the absolute Jacobian determinant.

∬Rf(x,y) dA=∬Sf(x(u,v),y(u,v))∣∂(x,y)∂(u,v)∣ du dv.\iint_R f(x,y)\,dA = \iint_S f(x(u,v),y(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| \,du\,dv.

2D Jacobian

Determinant of the derivative matrix.

∂(x,y)∂(u,v)=∣xuxvyuyv∣=xuyv−xvyu.\frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} x_u & x_v\\ y_u & y_v \end{vmatrix} = x_u y_v-x_v y_u.

Transform the whole problem

A correct change of variables requires all four pieces: region, bounds, integrand, and differential element. Computing a Jacobian alone is not a complete transformation.

Three-Dimensional Theorem

3D Change of Variables

Let T(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w))T(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w)) be a suitable one-to-one continuously differentiable transformation from SS to E=T(S)E=T(S). Then

3D Change-of-Variables Theorem

Transform a triple integral using the absolute 3D Jacobian determinant.

∭Ef(x,y,z) dV=∭Sf(T(u,v,w))∣∂(x,y,z)∂(u,v,w)∣ du dv dw.\iiint_E f(x,y,z)\,dV = \iiint_S f(T(u,v,w)) \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| \,du\,dv\,dw.

3D Jacobian

Determinant of the 3 by 3 derivative matrix.

∂(x,y,z)∂(u,v,w)=det⁡(xuxvxwyuyvywzuzvzw).\frac{\partial(x,y,z)}{\partial(u,v,w)} = \det \begin{pmatrix} x_u & x_v & x_w\\ y_u & y_v & y_w\\ z_u & z_v & z_w \end{pmatrix}.

Familiar coordinate systems are change-of-variables formulas

  • Polar coordinates have ∣∂(x,y)/∂(r,θ)∣=r\left|\partial(x,y)/\partial(r,\theta)\right|=r.
  • Cylindrical coordinates have ∣∂(x,y,z)/∂(r,θ,z)∣=r\left|\partial(x,y,z)/\partial(r,\theta,z)\right|=r.
  • Spherical coordinates have ∣∂(x,y,z)/∂(ρ,ϕ,θ)∣=ρ2sin⁡ϕ\left|\partial(x,y,z)/\partial(\rho,\phi,\theta)\right|=\rho^2\sin\phi under the convention used here.

Bounds under a General Transformation

If the standard axes are awkward, transform the entire problem together. This route checks region data, coordinate coverage, the forward Jacobian, and the final bounds before evaluation; unresolved geometry remains a hold rather than an invented limit.

Triple-Integral Bounds and Coordinate Transformation

Specify the solid, choose coordinates that simplify the geometry and integrand, transform the complete integral, and verify one-time coverage

Triple-Integral Bounds and Coordinate TransformationSpecify the solid, choose coordinates that simplify the geometry and integrand, transform the complete integral, and verify one-time coverage. Start with solid E and field f → Record E, f and the requested quantity; Record E, f and the requested quantity → Is region data sufficient for bounds?; Is region data sufficient for bounds? — No → Resolve missing solid or field data; Is region data sufficient for bounds? — Yes → Are rectangular bounds simple?; Project E to set rectangular bounds → Do the slices cover E exactly once?; Do the slices cover E exactly once? — Yes → Evaluate and report bounds and order; Do the slices cover E exactly once? — No → Split E into pieces with single-valued limits; Split E into pieces with single-valued limits → Do the pieces cover E exactly once?; Do the pieces cover E exactly once? — Yes → Sum the piece integrals; report bounds; Do the pieces cover E exactly once? — No → Hold: resolve gaps or overlaps; Transform E and f; include r or ρ² sin φ → Do radial/angular bounds cover E once?; Do radial/angular bounds cover E once? — Yes → Evaluate; report coordinates and bounds; Do radial/angular bounds cover E once? — No → Hold: revise radial or angular bounds; Define T:S→E and compute |det DT| → Are T, Jacobian and coverage valid?; Are T, Jacobian and coverage valid? — Yes → Integrate (f ∘ T)|det DT| over S; Are T, Jacobian and coverage valid? — No → Hold: verify the map and coverage; Are rectangular bounds simple? — Yes → Project E to set rectangular bounds; Are rectangular bounds simple? — No → Do cylindrical or spherical coordinates simplify both?; Do cylindrical or spherical coordinates simplify both? — Yes → Transform E and f; include r or ρ² sin φ; Do cylindrical or spherical coordinates simplify both? — No → Define T:S→E and compute |det DT|

Start with solid E and field f → Record E, f and the requested quantity; Record E, f and the requested quantity → Is region data sufficient for bounds?; Is region data sufficient for bounds? — No → Resolve missing solid or field data; Is region data sufficient for bounds? — Yes → Are rectangular bounds simple?; Project E to set rectangular bounds → Do the slices cover E exactly once?; Do the slices cover E exactly once? — Yes → Evaluate and report bounds and order; Do the slices cover E exactly once? — No → Split E into pieces with single-valued limits; Split E into pieces with single-valued limits → Do the pieces cover E exactly once?; Do the pieces cover E exactly once? — Yes → Sum the piece integrals; report bounds; Do the pieces cover E exactly once? — No → Hold: resolve gaps or overlaps; Transform E and f; include r or ρ² sin φ → Do radial/angular bounds cover E once?; Do radial/angular bounds cover E once? — Yes → Evaluate; report coordinates and bounds; Do radial/angular bounds cover E once? — No → Hold: revise radial or angular bounds; Define T:S→E and compute |det DT| → Are T, Jacobian and coverage valid?; Are T, Jacobian and coverage valid? — Yes → Integrate (f ∘ T)|det DT| over S; Are T, Jacobian and coverage valid? — No → Hold: verify the map and coverage; Are rectangular bounds simple? — Yes → Project E to set rectangular bounds; Are rectangular bounds simple? — No → Do cylindrical or spherical coordinates simplify both?; Do cylindrical or spherical coordinates simplify both? — Yes → Transform E and f; include r or ρ² sin φ; Do cylindrical or spherical coordinates simplify both? — No → Define T:S→E and compute |det DT|

  • Start with solid E and field f: terminator
  • Record E, f and the requested quantity: input output
  • Is region data sufficient for bounds?: decision
  • Resolve missing solid or field data: document
  • Are rectangular bounds simple?: decision
  • Project E to set rectangular bounds: process. Choose an integration order, project E onto the outer-coordinate plane, and write the iterated limits for each slice.
  • Do the slices cover E exactly once?: decision
  • Split E into pieces with single-valued limits: process. Make the pieces nonoverlapping before integrating; then verify their combined coverage.
  • Evaluate and report bounds and order: document
  • Do the pieces cover E exactly once?: decision
  • Hold: resolve gaps or overlaps: document
  • Sum the piece integrals; report bounds: document
  • Evaluate; report coordinates and bounds: document
  • Hold: revise radial or angular bounds: document
  • Hold: verify the map and coverage: document. Check one-to-one regular patches and valid bounds, or choose a different transformation.
  • Integrate (f ∘ T)|det DT| over S: document. State the map, transformed region, bounds, and complete integral setup.
  • Transform E and f; include r or ρ² sin φ: process. Compare cylindrical and spherical coordinates, transform the boundaries and integrand, and use the matching volume Jacobian.
  • Do radial/angular bounds cover E once?: decision
  • Define T:S→E and compute |det DT|: process. Transform the surfaces, region, and integrand using the forward Jacobian determinant.
  • Are T, Jacobian and coverage valid?: decision
  • Do cylindrical or spherical coordinates simplify both?: decision

General Transformation Workflow

  • Identify combinations of x,y,zx,y,z that repeatedly appear in the region or integrand.
  • Define new variables so those combinations become simple coordinate lines, planes, or constants.
  • Solve for the old variables in terms of the new variables when practical.
  • Transform every boundary equation and determine the new region.
  • Compute the correct forward Jacobian ∂(x,y)/∂(u,v)\partial(x,y)/\partial(u,v) or ∂(x,y,z)/∂(u,v,w)\partial(x,y,z)/\partial(u,v,w), or use the reciprocal of the inverse Jacobian where valid.
  • Insert the absolute value of the Jacobian determinant.
  • Evaluate and check units, sign, and order of magnitude.

Forward versus inverse Jacobian

If you compute ∂(u,v)/∂(x,y)\partial(u,v)/\partial(x,y) instead, then where the transformation is locally invertible,

∣∂(x,y)∂(u,v)∣=1∣∂(u,v)∂(x,y)∣.\left| \frac{\partial(x,y)}{\partial(u,v)} \right| = \frac{1}{ \left| \frac{\partial(u,v)}{\partial(x,y)} \right| }.

Do not use the inverse determinant without taking the reciprocal.

The absolute Jacobian preserves positive area under orientation reversal

The map T(u,v) = (2u + v, u − v) has determinant J = −3. It reverses orientation, but sends a unit square of area 1 to a parallelogram of area 3, so the area factor is |J| = 3.

A unit square maps to a parallelogram three times its area under a linear transformation with Jacobian minus three; the ordinary area factor is the absolute value three.
Key Takeaways
  • A triple integral accumulates a scalar field over ordinary three-dimensional volume; ∭E1 dV\iiint_E 1\,dV is the volume of EE.
  • Cylindrical coordinates use dV=r dz dr dθdV=r\,dz\,dr\,d\theta and spherical coordinates use dV=ρ2sin⁡ϕ dρ dϕ dθdV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta under this topic's angle convention.
  • Mass, center of mass, and inertia follow from integrating density times the appropriate lever-arm factors.
  • Coordinate choice should simplify the region and integrand together.
  • The change-of-variables theorem transforms the region, integrand, bounds, and differential element and uses the absolute Jacobian determinant.
  • General two- and three-dimensional Jacobians extend the same scaling idea seen in polar, cylindrical, and spherical coordinates.

References