Triple Integrals and Change of Variables - Theory & Concepts

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 $E$ . It does not require a four-dimensional geometric interpretation: $\iiint_E 1\,dV$ is the ordinary three-dimensional volume of $E$ , while $\iiint_E f\,dV$ accumulates the scalar field $f$ over that volume.

Triple Integrals over Solids

$Triple Integral$

For an integrable scalar field $f(x,y,z)$ over a solid $E\subset\mathbb{R}^3$ , the triple integral $\iiint_E f\,dV$ is the limit of sums of the form $\sum f(x_i^*,y_i^*,z_i^*)\,\Delta V$ .

Triple Integral

Scalar accumulation over a three-dimensional solid.

\iiint_E f(x,y,z)\,dV.

$Geometric and physical meanings$

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

Rectangular Coordinates and Bounds Construction

For the order $dz\,dy\,dx$ , first project the solid onto the $xy$ -plane. Describe that projection by $a\le x\le b$ and $g_1(x)\le y\le g_2(x)$ . Then, at each point $(x,y)$ in the projection, describe the lower and upper surfaces in the $z$ -direction.

Typical dz dy dx Setup

A solid bounded vertically by two surfaces.

\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.

$Constructing Triple-Integral Bounds$

STEP-BY-STEP

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.

Cylindrical Coordinates

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

$Cylindrical Coordinate Relations$

$x=r\cos\theta$
$y=r\sin\theta$
$z=z$
$x^2+y^2=r^2$
$r\ge0$

Cylindrical Volume Element

Jacobian-scaled volume differential.

dV=r\,dz\,dr\,d\theta

for the displayed order. Any consistent permutation of $dz$ , $dr$ , and $d\theta$ is possible when the corresponding bounds are rewritten correctly.

$The factor r is mandatory$

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

Spherical Coordinates

This course uses $\rho$ for distance from the origin, $\theta$ for the azimuth angle in the $xy$ -plane, and $\phi$ for the polar angle measured from the positive $z$ -axis.

$Spherical Coordinate Relations$

$x=\rho\sin\phi\cos\theta$
$y=\rho\sin\phi\sin\theta$
$z=\rho\cos\phi$
$x^2+y^2+z^2=\rho^2$
$r=\rho\sin\phi$
$\rho\ge0$ , usually $0\le\theta\le2\pi$ , $0\le\phi\le\pi$

Spherical Volume Element

Jacobian-scaled volume differential.

dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta

$Angle convention$

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

Coordinate-Selection Strategy

$Checklist$

Use rectangular coordinates when bounding planes and coordinate-aligned boxes dominate.
Use cylindrical coordinates when $x^2+y^2$ , circular projections, cones about the $z$ -axis, or surfaces $z=g(r)$ dominate.
Use spherical coordinates when spheres centered at the origin, cones $\phi=\text{constant}$ , or radial expressions $x^2+y^2+z^2$ 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 $\delta(x,y,z)\ge0$ denote volumetric mass density.

Mass of a Solid

Integral of volumetric density.

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

First Moments

Moments about the coordinate planes.

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.

\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.

I_x=\iiint_E(y^2+z^2)\delta\,dV,

I_y=\iiint_E(x^2+z^2)\delta\,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 $T(u,v)=(x(u,v),y(u,v))$ be a continuously differentiable one-to-one transformation on a region $S$ (up to standard boundary exceptions), with nonzero Jacobian in the interior. If $R=T(S)$ , then

2D Change-of-Variables Theorem

Transform a double integral using the absolute Jacobian determinant.

\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.

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

3D Change-of-Variables Theorem

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

\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.

\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 $\left|\partial(x,y)/\partial(r,\theta)\right|=r$ .
Cylindrical coordinates have $\left|\partial(x,y,z)/\partial(r,\theta,z)\right|=r$ .
Spherical coordinates have $\left|\partial(x,y,z)/\partial(\rho,\phi,\theta)\right|=\rho^2\sin\phi$ under the convention used here.

Bounds under a General Transformation

$General Transformation Workflow$

STEP-BY-STEP

Identify combinations of $x,y,z$ 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 $\partial(x,y)/\partial(u,v)$ or $\partial(x,y,z)/\partial(u,v,w)$ , 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 $\partial(u,v)/\partial(x,y)$ instead, then where the transformation is locally invertible,

\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.

Key Takeaways

A triple integral accumulates a scalar field over ordinary three-dimensional volume; $\iiint_E 1\,dV$ is the volume of $E$ .
Cylindrical coordinates use $dV=r\,dz\,dr\,d\theta$ and spherical coordinates use $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ 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.

Previous TopicMultiple Integrals - Examples & Applications

Quiz Me

Next TopicTriple Integrals and Change of Variables - Examples & Applications

Prev Next

Quiz Me