A student sets up the volume integral for the solid cylinder r ≤ 2, 0 ≤ z ≤ 3 in cylindrical coordinates and writes ∫∫∫ dr dθ dz over the appropriate limits. What error did they make?
AThey should use spherical coordinates for cylindrical solids, not cylindrical coordinates
BThe z-limits should be centered at zero, from −3/2 to 3/2
CThey omitted the factor of r — the volume element is r dr dθ dz, not dr dθ dz
DThe order of integration must always be dz dr dθ in cylindrical coordinates
The factor of r is mandatory in the cylindrical volume element. A thin wedge at radius r with angular opening dθ has arc length r dθ in the tangential direction — not just dθ. Without r, you treat all wedges as having the same arc width regardless of radius, which overcounts near the origin and undercounts far from it. The error produces a volume that is systematically wrong. This is the same Jacobian factor that appears in polar area elements: dA = r dr dθ.
Question 2 Multiple Choice
Which of the following integrals would be MOST efficiently simplified by switching to cylindrical coordinates?
A∫∫∫ (x² + y² + z²) dV over a sphere — use spherical coordinates instead
B∫∫∫ e^(x² + y²) dV over the region x² + y² ≤ 4, 0 ≤ z ≤ 1
C∫∫∫ xyz dV over the rectangular box [0,1] × [0,1] × [0,1]
D∫∫∫ z² dV over the unit sphere — use spherical coordinates instead
The integrand e^(x² + y²) contains x² + y², which collapses to r² in cylindrical coordinates: the integral becomes ∫∫∫ e^(r²) r dr dθ dz, which is separable. The circular region of integration x² + y² ≤ 4 becomes r ≤ 2, a natural cylindrical boundary. When the integrand or boundary involves x² + y², that is the telltale sign to use cylindrical. When it involves x² + y² + z², use spherical instead.
Question 3 True / False
Cylindrical coordinates introduce fundamentally new mathematics compared to polar coordinates — they require learning separate conversion formulas and a different integration framework.
TTrue
FFalse
Answer: False
Cylindrical coordinates extend polar coordinates to three dimensions by simply appending an unchanged z-coordinate. The conversion formulas x = r cosθ, y = r sinθ, z = z are exactly the polar formulas with z added. The volume element r dr dθ dz comes directly from the polar area element r dr dθ with dz appended. Students who understand polar coordinates can grasp cylindrical coordinates immediately — there are no genuinely new ideas, only a straightforward extension.
Question 4 True / False
In the cylindrical volume element r dr dθ dz, the factor r appears because arc length in the angular direction equals r times the angular increment dθ.
TTrue
FFalse
Answer: True
This is the geometric origin of the Jacobian factor. A thin wedge at radius r spanning an angular increment dθ has arc length r dθ at that radius — the further from the axis, the wider the wedge becomes in absolute terms. The volume of the wedge element is (radial extent dr) × (arc extent r dθ) × (vertical extent dz) = r dr dθ dz. The same reasoning gives the polar area element r dr dθ. The r factor is not optional; it reflects real geometry.
Question 5 Short Answer
Why does the volume element in cylindrical coordinates include a factor of r, and what goes wrong in a volume calculation if you omit it?
Think about your answer, then reveal below.
Model answer: The r factor arises because the angular coordinate θ does not directly measure length — arc length in the θ direction at radius r is r dθ, not dθ. So a tiny volume wedge has dimensions dr (radial) × r dθ (arc) × dz (vertical) = r dr dθ dz. Without the r factor, all wedges at different radii are treated as having the same physical width, which is false: wedges near the origin are narrow, and those far away are wide. Omitting r overcounts volume near the axis and undercounts it far from the axis, producing an incorrect result.
This is the single most important fact about integration in cylindrical coordinates. The Jacobian factor r is what connects the coordinate increment dθ to actual physical arc length — it is not an algebraic technicality but a geometric necessity. The same principle explains why polar area integrals use r dr dθ.