Volume of a cylinder in cylindrical coordinates
∫₀²ᵖⁱ∫₀¹∫₀³ r dz dr dθ = 3π ≈ 9.42477796. A worked triple integral in cylindrical coordinates, with an editable calculator preloaded with the limits.
∫₀²ᵖⁱ∫₀¹∫₀³ r dz dr dθ = 3π ≈ 9.42477796. The Jacobian r is what turns dz dr dθ into a true volume element.
dV = r dz dr dθ — the extra r is added for you
In terms of θ, r, z
Limits of integration
Inner limits may use the outer variables — e.g. 1-x, sqrt(1-x^2)
The setup
∫₀²ᵖⁱ∫₀¹∫₀³ r dz dr dθ = 3π ≈ 9.42477796
The Jacobian r is what turns dz dr dθ into a true volume element.
Coordinate system: cylindrical. The integrand is 1, integrated from 0 to 3 innermost, then 0 to 1, then 0 to 2*pi.
More worked triple integrals
Frequently asked questions
What is the value of ∫₀²ᵖⁱ∫₀¹∫₀³ r dz dr dθ? ▾
It evaluates to 3π ≈ 9.42477796. The Jacobian r is what turns dz dr dθ into a true volume element.
Why cylindrical coordinates for this one? ▾
The region is symmetric about the z-axis, so circular cross-sections become the simple limits 0 ≤ r ≤ R once you switch to cylindrical coordinates.
Do I need to include the Jacobian myself? ▾
No. Enter only your integrand; the calculator multiplies by r for you.
About this Calculator
∫₀²ᵖⁱ∫₀¹∫₀³ r dz dr dθ = 3π ≈ 9.42477796. A worked triple integral in cylindrical coordinates, with an editable calculator preloaded with the limits.