Checked example
Worked calculation
Result: With r = 2 and h = 5, ∫₀⁵ 4π dz = 20π ≈ 62.8319 cubic units.
01 • Guide
Cross-sectional method
Choose an axis perpendicular to the circular base. A thin slice of thickness dz has approximate volume πr² dz. The definite integral is the limit of the sum of all such slices.
Keep the full precision of intermediate values and round only the final display. This avoids compounding error when a derived radius or converted unit is used in another step.
02 • Guide
Units inside the integral
The integrand is an area measured in square units, while dz supplies a length. Their product is a tiny volume, and integration adds those pieces into cubic units.
The calculator validates physical limits before displaying a result. Blank or impossible inputs never produce a stale value, NaN or Infinity.
03 • Guide
How the method generalises
If radius changes with position, replace r with r(z). Cones, frustums and curved vessels use ∫π[r(z)]²dz. A true cylinder is the especially simple constant-radius case.
Interpretation
How to use the result responsibly
The answer models an ideal geometric cylinder. Manufacturing tolerances, wall thickness, rounded ends, deformation, tank tilt and measurement uncertainty can all shift a real value. Use inside dimensions for capacity and outside dimensions for occupied space. For safety, compliance, dosing, structural or purchasing decisions, compare the estimate with drawings, calibration data or a qualified professional.
Dimensional checks help catch mistakes: a volume result must carry cubic units, an area result square units and a derived radius or height linear units. If changing one unit alters the represented physical size, a conversion step was missed.