Questions: Triple Integrals in Cylindrical Coordinates

Question 1

A student computes ∫₀²π ∫₀ᴿ ∫₀ᴴ f(r,θ,z) dz dr dθ to find the mass of a cylinder. What critical error have they made?

• A. The order of integration is wrong — z must be the outermost integral
• B. The bounds on θ should go from 0 to π, not 0 to 2π
• C. The volume element is missing the factor r — the correct element is r dz dr dθ, giving an extra factor that grows with radius
• D. f must be expressed in Cartesian coordinates before integration

Answer: (C)

The volume element in cylindrical coordinates is dV = r dr dθ dz (in any order of integration). The student wrote dz dr dθ, omitting the essential factor of r. This factor arises from the Jacobian of the coordinate transformation: a change of angle dθ sweeps an arc of length r dθ, not dθ, so the infinitesimal 'width' of the volume element in the θ-direction is r dθ. Omitting r systematically undercounts volume, with the error proportional to how far from the z-axis the integration occurs. The order of integration (option A) can be rearranged without affecting correctness, and option B gives the wrong angular range for a full cylinder.

Question 2

Which of the following regions is BEST suited for cylindrical coordinates rather than Cartesian?

• A. A rectangular box 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 5
• B. The region inside the sphere x² + y² + z² = 4 and above the plane z = 1
• C. The solid bounded below by z = 0, above by z = 4 − r², and inside the cylinder x² + y² = 1
• D. The tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), (0,0,1)

Answer: (C)

Cylindrical coordinates excel when the region has rotational symmetry around the z-axis, which converts curved Cartesian boundaries into simple inequalities on r and z. The region bounded by z = 4 − r² (a paraboloid in cylindrical form) and the cylinder x² + y² = 1 (which becomes simply r = 1) is naturally described as 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 4 − r². In Cartesian, the paraboloid z = 4 − x² − y² would require dealing with a circular cross-section producing square roots. Options A and D are natural for Cartesian. Option B has spherical symmetry, making spherical coordinates the better choice.

Question 3

The extra factor of r in the cylindrical volume element dV = r dr dθ dz arises because a small change in angle dθ sweeps an arc of physical length r dθ, not dθ.

• T. True
• F. False

Answer: True

This is the geometric heart of the result. In polar/cylindrical coordinates, equal angular increments dθ correspond to arcs of different physical lengths depending on radius: the arc length is r dθ. A small 'box' in cylindrical coordinates has dimensions dr in the radial direction, r dθ in the angular direction, and dz vertically, giving volume dr × r dθ × dz = r dr dθ dz. The Jacobian of the transformation formalizes this: |∂(x,y,z)/∂(r,θ,z)| = r. Farther from the z-axis, the same angular increment sweeps more physical space, which is why r appears as a multiplicative factor.

Question 4

The volume element in cylindrical coordinates is dr dθ dz, because θ is dimensionless (measured in radians) and contributes no length factor.

• T. True
• F. False

Answer: False

This is the most common error when switching to cylindrical coordinates. Radians are indeed dimensionless, but that is beside the point: a radian is a ratio of arc length to radius, so an arc of angle dθ has physical length r dθ. Dimensionlessness of the angle does not mean the arc it sweeps has zero extent. The correct volume element is dV = r dr dθ dz, with r as the Jacobian factor. Forgetting r produces integrals that systematically undercount volume, with the error largest for regions at large r.

Question 5

Why does the volume element in cylindrical coordinates include a factor of r, and what goes wrong computationally if you forget it?

Model answer: The factor r is the Jacobian of the transformation from Cartesian (x,y,z) to cylindrical (r,θ,z). It accounts for the fact that equal increments of angle θ sweep different amounts of physical space depending on the radius: the arc length is r dθ. An infinitesimal cylindrical 'box' has radial width dr, circumferential width r dθ, and height dz, giving volume r dr dθ dz. If you omit r and use dr dθ dz, you are treating angular increments as having unit width everywhere — undercounting volume in proportion to how far from the axis the region lies. For a symmetric region like a cylinder, the error factor is exactly ∫₀ᴿ r dr / ∫₀ᴿ dr = R/2, so you undercount by half the radius on average.

The Jacobian is the formal machinery behind this: changing variables in a multiple integral requires multiplying by |det(J)| where J is the matrix of partial derivatives of the new coordinates with respect to the old. For cylindrical coordinates, |det(J)| = r. The practical diagnostic: if your answer for the volume of a cylinder of radius R and height H is πR²H but your integral gives RH (missing the R factor), you forgot the r in dV.