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Triple Integrals in Cylindrical and Spherical Coordinates cheat sheet - grade college

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Calculus Grade college

Triple Integrals in Cylindrical and Spherical Coordinates Cheat Sheet

A printable reference covering cylindrical coordinates, spherical coordinates, Jacobian factors, volume elements, and triple integral setup for college.

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Triple integrals measure volume, mass, charge, probability, and other quantities spread throughout three-dimensional regions. Cylindrical and spherical coordinates make many integrals easier when a region has circular, radial, or spherical symmetry. This cheat sheet helps students choose coordinates, convert formulas, and set correct bounds.

It is especially useful for setting up integrals before doing the calculation.

Cylindrical coordinates use x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta, z=zz = z, and the volume element dV=r dz dr dθdV = r\,dz\,dr\,d\theta. Spherical coordinates use x=ρsin⁡ϕcos⁡θx = \rho\sin\phi\cos\theta, y=ρsin⁡ϕsin⁡θy = \rho\sin\phi\sin\theta, z=ρcos⁡ϕz = \rho\cos\phi, and dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta. The extra factors rr and ρ2sin⁡ϕ\rho^2\sin\phi come from the Jacobian of the coordinate change.

Good setup means matching the coordinate system to the region, rewriting the integrand, and choosing bounds that describe the solid exactly once.

Key Facts

  • Cylindrical coordinates are defined by x=rcos⁡θx = r\cos\theta, y=rsin⁡θy = r\sin\theta, z=zz = z, with r≥0r \ge 0.
  • The cylindrical volume element is dV=r dz dr dθdV = r\,dz\,dr\,d\theta, so every cylindrical triple integral must include the factor rr.
  • Spherical coordinates are defined by x=ρsin⁡ϕcos⁡θx = \rho\sin\phi\cos\theta, y=ρsin⁡ϕsin⁡θy = \rho\sin\phi\sin\theta, and z=ρcos⁡ϕz = \rho\cos\phi, with ρ≥0\rho \ge 0.
  • The spherical volume element is dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta, so every spherical triple integral must include ρ2sin⁡ϕ\rho^2\sin\phi.
  • The common angle convention in calculus is 0≤θ≤2π0 \le \theta \le 2\pi around the zz-axis and 0≤ϕ≤π0 \le \phi \le \pi down from the positive zz-axis.
  • Useful conversions include r2=x2+y2r^2 = x^2 + y^2, ρ2=x2+y2+z2\rho^2 = x^2 + y^2 + z^2, z=ρcos⁡ϕz = \rho\cos\phi, and r=ρsin⁡ϕr = \rho\sin\phi.
  • Cylindrical coordinates are usually best for cylinders, cones, and solids of revolution around the zz-axis.
  • Spherical coordinates are usually best for spheres, balls, spherical shells, and cones with vertex at the origin.

Vocabulary

Cylindrical coordinates
A coordinate system using distance from the zz-axis, angle around the zz-axis, and height, written as (r,θ,z)(r,\theta,z).
Spherical coordinates
A coordinate system using distance from the origin and two angles, written as (ρ,θ,ϕ)(\rho,\theta,\phi).
Jacobian
The scaling factor that adjusts area or volume when changing variables in an integral.
Volume element
The differential piece of volume, such as dV=r dz dr dθdV = r\,dz\,dr\,d\theta or dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta.
Azimuthal angle
The angle θ\theta measured in the xyxy-plane around the zz-axis.
Polar angle
The angle ϕ\phi measured from the positive zz-axis in spherical coordinates.

Common Mistakes to Avoid

  • Forgetting the Jacobian factor is wrong because dVdV is not just dz dr dθdz\,dr\,d\theta or dρ dϕ dθd\rho\,d\phi\,d\theta after a coordinate change. Use dV=r dz dr dθdV = r\,dz\,dr\,d\theta in cylindrical coordinates and dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta in spherical coordinates.
  • Confusing ρ\rho and rr is wrong because rr measures distance from the zz-axis while ρ\rho measures distance from the origin. Remember r=ρsin⁡ϕr = \rho\sin\phi and z=ρcos⁡ϕz = \rho\cos\phi.
  • Using the wrong angle meaning is wrong because θ\theta and ϕ\phi describe different rotations. In the standard convention, θ\theta goes around the zz-axis and ϕ\phi goes down from the positive zz-axis.
  • Leaving the integrand in rectangular variables is wrong because all variables must match the chosen coordinate system. Replace expressions such as x2+y2x^2 + y^2 with r2r^2 in cylindrical coordinates or ρ2sin⁡2ϕ\rho^2\sin^2\phi in spherical coordinates.
  • Setting bounds that trace the region more than once is wrong because the integral will overcount volume or mass. Use ranges such as 0≤θ≤2π0 \le \theta \le 2\pi for one full rotation unless the region covers only part of the circle.

Practice Questions

  1. 1 Set up and evaluate the volume of the solid cylinder x2+y2≤9x^2 + y^2 \le 9 with 0≤z≤40 \le z \le 4 using cylindrical coordinates.
  2. 2 Set up and evaluate the volume of the ball x2+y2+z2≤16x^2 + y^2 + z^2 \le 16 using spherical coordinates.
  3. 3 Rewrite the integral ∭E(x2+y2) dV\iiint_E (x^2 + y^2)\,dV in cylindrical coordinates for the region 0≤z≤5−x2−y20 \le z \le 5 - x^2 - y^2 above the xyxy-plane.
  4. 4 A solid is bounded by a sphere centered at the origin and a cone with vertex at the origin. Explain why spherical coordinates are usually a better choice than cylindrical coordinates.

Understanding Triple Integrals in Cylindrical and Spherical Coordinates

The scale factors are not optional decoration. They correct for the fact that equal changes in the new coordinates do not produce equal-sized boxes in space. Near the axis, a small turn through an angle sweeps out a narrow wedge.

Farther out, the same turn sweeps out a wider wedge. That is why cylindrical slices need a factor of r. In spherical coordinates, the size of a small patch changes with both distance from the origin and direction.

A patch near a pole is narrow because circles of latitude shrink there. A patch farther from the origin covers much more space.

The spherical factor accounts for both effects. Leaving out a scale factor means adding pieces that have been assigned the wrong volumes.

A strong setup begins by translating the boundary surfaces, not by choosing limits immediately. A vertical circular wall becomes a constant radial distance in cylindrical coordinates. A horizontal plane stays a statement about z.

A cone with its tip at the origin often becomes a constant spherical angle. A sphere centered at the origin becomes a constant distance from the origin. Sketching a cross section in the rz plane is especially useful.

It shows whether the upper and lower surfaces change as the radius changes. For example, a solid inside a sphere but above a cone may have simple spherical limits, while the same solid can require several linked expressions in rectangular coordinates.

The order of integration can change how hard the work becomes. Choose an order that makes each bound easy to state. In cylindrical coordinates, it is often convenient to let height vary first, then let radius move across the base, then sweep through the angle.

This works well when a solid sits between two surfaces described by height. In spherical coordinates, distance from the origin often varies first. The polar angle then selects a band from top to bottom, and the azimuthal angle sweeps around the axis.

A full rotation uses the complete angle range, but a half solid or quarter solid needs only the part actually present. Symmetry can reduce work when the integrand has matching behavior on opposite sides.

These coordinate systems appear in models of tanks, pipes, domes, planets, magnetic fields, heat flow, and density inside manufactured parts. A density may increase with height, depend only on distance from an axis, or depend only on distance from a center. The coordinate choice should match that pattern as well as the shape.

When studying, pay close attention to which angle is measured around the vertical axis and which angle is measured downward from the positive vertical direction. Mixing those angles is a common error. Another common error is describing a region more than once, especially when an angle range overlaps itself.

Before calculating, test a few points at the boundaries and state in words what each limit controls. That habit catches many setup mistakes before the algebra begins.