volume integral

Volume Integral: Definition, Geometry, Density, and Physical Applications

A volume integral extends integration to a three-dimensional region.

Its most basic form is

(1)

where:

• V is a three-dimensional region;
• f(r) is a scalar field defined throughout that region;
• dV is an infinitesimal volume element.

Volume integrals are fundamental because many physical quantities are distributed throughout space rather than concentrated on a line or surface.

Examples include:

• mass distributed through a solid body;
• Electric Charge distributed through a volume;
• internal or electromagnetic energy density;
• probability density in quantum mechanics;
• source density in field equations;
• moments of inertia and center of mass;
• total divergence inside a region.

A useful physical interpretation is

(2)

1 From sums to volume integrals

Divide a region V into many small subvolumes

(1)

Choose a point r_i in each subvolume.

A Riemann sum is

(2)

As the largest subvolume dimension approaches zero, the sum approaches the volume integral:

(3)

This definition makes clear that dV represents a geometrically small volume over which the field is approximately constant.

2 Volume itself as an integral

Setting

(3)

gives

(4)

Thus the geometric volume of a region is itself the simplest volume integral.

This parallels:

(4)

for curve length and

(5)

for surface area.

3 Cartesian coordinates

In Cartesian coordinates,

(6)

A small rectangular box has side lengths

(7)

Therefore

(5)

A Cartesian triple integral is

(6)

4 Iterated integrals

Under suitable regularity conditions, a triple integral can be evaluated as nested one-dimensional integrals.

For a rectangular box,

(8)

we may write

(7)

The order may often be changed, provided the bounds are transformed consistently.

For more complicated regions, one or more limits depend on the outer variables.

5 Example 1: volume under a paraboloid

Find the volume bounded by

(9)

and the plane

(10)

The intersection satisfies

(11)

Using cylindrical coordinates in the horizontal plane,

(12)

Then

𝒱	= ∫ 02π ∫ 02 ∫ 04−r2 r dz dr dϕ	(13)
	= 2π ∫ 02r(4 − r2) dr	(14)
	= 2π02	(15)
	= 8π.	(8)

This example already hints that the volume element changes when coordinates change.

6 Coordinate transformations and the Jacobian

Suppose coordinates (u,v,w) are related to Cartesian coordinates by

(16)

A small coordinate box

(17)

maps to a small parallelepiped in physical space.

The volume scaling is given by the Jacobian determinant:

(9)

The determinant is

(18)

The absolute value is used because a volume measure is nonnegative.

7 Geometric form of the Jacobian

Define the coordinate tangent vectors

(19)

The small parallelepiped volume is

(10)

The scalar triple product is exactly the Jacobian volume scaling.

8 Cylindrical coordinates

Use

x	= ρ cos ϕ,	(20)
y	= ρ sin ϕ,	(21)
z	= z.	(22)

The coordinate scale factors are:

• radial length dρ;
• azimuthal arc length ρdϕ;
• vertical length dz.

Therefore

(11)

The factor ρ is the cylindrical-coordinate Jacobian.

9 Example 2: volume of a cylinder

For a cylinder of radius R and height H,

0	≤ ρ ≤ R,	(23)
0	≤ ϕ < 2π,	(24)
0	≤ z ≤ H.	(25)

Hence

𝒱	= ∫ 0H ∫ 02π ∫ 0Rρdρdϕdz	(26)
	= H(2π)	(27)
	= πR2H.	(12)

10 Spherical coordinates

Use the convention

x	= r sin 𝜃 cos ϕ,	(28)
y	= r sin 𝜃 sin ϕ,	(29)
z	= r cos 𝜃,	(30)

where:

r	≥ 0,	(31)
0	≤ 𝜃 ≤ π,	(32)
0	≤ ϕ < 2π.	(33)

The local side lengths are approximately

dr,		(34)
r d𝜃,		(35)
r sin 𝜃 dϕ.		(36)

Therefore

(13)

11 Example 3: volume of a sphere

For a sphere of radius R,

0	≤ r ≤ R,	(37)
0	≤ 𝜃 ≤ π,	(38)
0	≤ ϕ < 2π.	(39)

Then

𝒱	= ∫ 02π ∫ 0π ∫ 0Rr2 sin 𝜃 dr d𝜃 dϕ	(40)
	= .	(41)

Thus

(14)

The familiar sphere-volume formula is therefore a direct triple-integral result.

12 Mass from volume density

Let the mass density be

(42)

A small volume contains mass

(43)

Therefore the total mass is

(15)

For constant density ρ₀,

(44)

13 Example 4: nonuniform spherical mass distribution

Suppose a sphere of radius R has density

(45)

The mass is

M	= ∫ 02π ∫ 0π ∫ 0Rρ 0r2 sin 𝜃 dr d𝜃 dϕ	(46)
	= 4πρ0 ∫ 0Rdr	(47)
	= 4πρ0.	(48)

Hence

(16)

The density vanishes at the surface and reaches ρ₀ at the center.

14 Average value over a volume

The average value of a scalar field f over a region of volume 𝒱 is

(17)

For the density in Example 4,

(18)

15 Center of mass

For a continuous body,

(19)

Componentwise,

xCM	= V xρm dV,	(49)
yCM	= V yρm dV,	(50)
zCM	= V zρm dV.	(51)

Symmetry can eliminate many of these integrals immediately.

16 Example 5: center of mass of a vertically graded block

Consider the rectangular block

(52)

with density

(53)

where α > −1.

By symmetry,

(54)

The mass is

M	= abρ0 ∫ 0Hdz	(55)
	= abρ0H.	(56)

The vertical moment is

V zρm dV	= abρ0 ∫ 0Hzdz	(57)
	= abρ0H2.	(58)

Therefore

(20)

For α > 0, the density increases upward and the center of mass lies above H∕2.

17 Moment of inertia

For rotation about a chosen axis, each mass element contributes

(59)

where r_⊥ is the perpendicular distance to the axis.

Since

(60)

the moment of inertia is

(21)

This is the continuum version of

(61)

18 Example 6: moment of inertia of a uniform solid sphere

Let the sphere have radius R, mass M, and uniform density

(62)

Take the rotation axis to be the z axis.

In spherical coordinates,

(63)

Thus

Iz	= r2 sin 2𝜃ρ 0r2 sin 𝜃 dr d𝜃 dϕ	(64)
	= ρ0.	(65)

Using

(66)

and

(67)

we obtain

Iz	= ρ02π	(68)
	= ρ0R5.	(69)

Substitute the uniform density:

(22)

19 Electric charge from volume charge density

Let

(70)

be a volume charge density.

Then

(71)

and the total charge is

(23)

This source integral appears directly in Gauss’s Law through

(72)

Then

(24)

The volume integral supplies the enclosed source; the surface integral measures its outward electric flux.

20 Energy density

If a physical system has energy density

(73)

then the total energy is

(25)

For the electromagnetic field in vacuum,

(74)

Thus

(26)

21 Example 7: energy stored in a parallel-plate capacitor

Consider an ideal parallel-plate capacitor with plate area A, separation d, and approximately uniform electric field E between the plates.

The electric-energy density is

(75)

The field-filled volume is

(76)

Therefore

(77)

Since

(78)

we obtain

U	= 𝜖0Ad	(79)
	= V 2.	(80)

For an ideal parallel-plate capacitor,

(81)

Hence

(27)

The familiar circuit formula is therefore also a volume integral of field-energy density.

22 Probability density in quantum mechanics

For a normalized three-dimensional wavefunction,

(82)

the probability density is

(83)

The probability of finding the particle in region V is

(28)

Normalization requires

(29)

Thus the same mathematical structure used for mass density and charge density also appears in quantum probability.

23 Vector-valued volume integrals

A vector field can also be integrated over a volume:

(30)

This is understood componentwise:

(84)

The center of mass formula is an important example because it integrates the vector position r weighted by scalar mass density.

24 The divergence theorem as a bridge between volume and surface integrals

The divergence theorem states

(31)

It converts a flux through the boundary into a volume integral of the local source-like quantity

(85)

This theorem is one of the deepest connections between the surface-integral and volume-integral articles.

25 Example 8: divergence theorem on a sphere

Let

(86)

Then

(87)

For a sphere of radius R,

V ∇⋅ FdV	= 3	(88)
	= 4πR3.	(32)

On the spherical boundary,

(89)

so

∮ ∂V F ⋅ dA	= R(4πR2)	(90)
	= 4πR3.	(33)

The volume and surface calculations agree exactly.

26 Local conservation laws

Many conservation laws can be written in differential form as

(34)

Here:

• ρ is a volume density of a conserved or tracked quantity;
• J is its flux;
• s is a local source term.

Integrate over a fixed volume:

(91)

Using the divergence theorem,

(35)

This says:

(92)

This structure appears in mass conservation, charge conservation, energy conservation, fluid dynamics, and transport theory.

27 Choosing the best coordinates

Coordinate choice is often the difference between a simple integral and a difficult one.

Use Cartesian coordinates when:

• the region is box-like;
• boundaries are planes aligned with coordinate axes;
• the integrand is simple in x, y, and z.

Use cylindrical coordinates when:

• the geometry is rotationally symmetric about an axis;
• circles or cylinders appear naturally;
• the integrand depends on x² + y².

Use spherical coordinates when:

• the geometry is centered on a point;
• spheres or cones appear;
• the integrand depends primarily on r.

A coordinate system should match the symmetry of both the region and the field whenever possible.

28 Dimensional checks

A volume integral should have dimensions equal to the integrand times volume.

For mass density,

(93)

so

(94)

For energy density,

(95)

so

(96)

For charge density,

(97)

so

(98)

Dimensional analysis is a powerful way to catch missing Jacobian factors.

29 Common mistakes

1. Using dxdy dz after changing to cylindrical or spherical coordinates.
2. Forgetting the cylindrical Jacobian factor ρ.
3. Forgetting the spherical Jacobian factor r² sin 𝜃.
4. Confusing density per unit volume with total quantity.
5. Integrating over the wrong geometric region.
6. Using constant bounds when the actual boundary depends on another coordinate.
7. Mixing the cylindrical radius ρ with a physical density symbol without defining notation clearly.
8. Forgetting that the Jacobian appears because coordinate cells have different physical volumes.
9. Treating center of mass as an unweighted geometric average when density is nonuniform.
10. Using total distance from the origin instead of perpendicular distance to the rotation axis in a moment-of-inertia integral.
11. Confusing a volume integral with a surface flux integral.
12. Applying the divergence theorem without a closed boundary.
13. Forgetting units and thereby missing a dimensional error.

30 Connections to other PhysicsLibrary articles

The volume integral connects directly to mass:

(99)

It connects to charge:

(100)

It connects to energy:

(101)

It connects to center of mass:

(102)

It connects to rotational mechanics:

(103)

It connects to the surface-integral article through

(104)

31 Summary

A volume integral adds a field throughout a three-dimensional region:

(105)

In Cartesian coordinates,

(106)

In cylindrical coordinates,

(107)

In spherical coordinates,

(108)

For a general coordinate transformation,

(109)

The central physical pattern is

(110)

This single structure produces mass, charge, energy, probability, center of mass, moments of inertia, and the volume side of the divergence theorem.

References

References