Calculator Input
Use variables r, theta, and z.
Supported functions include sin, cos, sqrt,
exp, log, min, and max.
Example Data Table
| Case | Integrand | θ limits | r limits | z limits | Meaning |
|---|---|---|---|---|---|
| Solid cylinder | 1 | 0 to 2*pi | 0 to 3 | 0 to 4 | Volume equals 36π with Jacobian applied. |
| Paraboloid cap | 1 | 0 to 2*pi | 0 to 2 | 0 to 5-r^2 | Finds volume below a curved surface. |
| Density field | r^2+z | 0 to pi | 0 to 2 | 0 to 5-r | Estimates mass for variable density. |
Formula Used
The calculator evaluates a cylindrical triple integral in this form:
∫[θ=a to b] ∫[r=g1(θ) to g2(θ)] ∫[z=h1(r,θ) to h2(r,θ)] f(r,θ,z) r dz dr dθ
The factor r is the cylindrical Jacobian. It converts the small box in
cylindrical coordinates into real volume. If your entered integrand already includes this
factor, clear the Jacobian checkbox.
Composite Simpson integration is applied through the z, r, and θ directions. Even panel counts improve smooth-function accuracy. Larger counts usually increase precision, but they also require more samples.
How to Use This Calculator
- Enter the function using
r,theta, andz. - Enter lower and upper limits for θ, r, and z.
- Use formulas in bounds when needed, such as
5-ror2+sin(theta). - Keep the Jacobian option checked for normal cylindrical integration.
- Choose even Simpson panel counts for each direction.
- Press calculate, then export the result as CSV or PDF.
Understanding Cylindrical Triple Integrals
Why Cylindrical Coordinates Help
Cylindrical coordinates describe a point by radius, angle, and height. They are useful when a region wraps around an axis. Many cylinders, pipes, tanks, cones, and circular caps become easier to describe this way. Instead of using x and y, the calculator uses r and theta. The height direction remains z.
Role of the Radius Factor
The most important detail is the radius factor. A thin slice near the outer edge covers more area than a thin slice near the center. The Jacobian handles that change. For ordinary volume, mass, and moment calculations, the integrand must be multiplied by r. This page can apply that factor automatically.
Variable Limits
Advanced regions often need variable limits. The radial upper limit may depend on theta. The height may depend on both r and theta. This calculator accepts those expressions. That makes it suitable for cylinders with sloped tops, circular sectors, offset density models, and curved caps.
Numerical Method
The tool uses nested Simpson rules. It first samples the z direction. Then it integrates the radial direction. Finally, it integrates across the angular direction. Simpson panels should be even. Smooth functions usually converge quickly. Sharp corners, discontinuities, and very steep surfaces need more panels.
Reading the Output
The main result is the requested integral. When the integrand is a density, the result is mass. When the integrand is one, the result is volume. The calculator also estimates geometric volume and a mean value. The mean value equals the integral divided by region volume. It helps compare different regions.
Practical Advice
Start with moderate panel counts. Then double them and compare results. If the values barely change, the estimate is stable. Keep units consistent across radius and height. Use radians unless you choose the degree option. For exact symbolic answers, use algebra software. For fast engineering estimates, this calculator is more direct.
FAQs
What is a cylindrical triple integral?
It is a triple integral written with radius, angle, and height. It is best for circular or rotational regions. The integral usually includes a radius multiplier because cylindrical volume elements expand outward.
Why is the Jacobian equal to r?
The arc length for a small angle is proportional to radius. A small angular slice becomes wider as r grows. The Jacobian r corrects the volume element.
Can I enter variable bounds?
Yes. The r bounds may use theta. The z bounds may use r and theta. Use expressions like 2+sin(theta), 5-r, or sqrt(9-r^2).
Should I use degrees or radians?
Radians are standard for calculus. Use degrees only when your entered θ limits are degree values. Trigonometric functions inside formulas still expect radians.
What Simpson panel count should I choose?
Use even counts. Twelve or twenty panels work for many smooth examples. Increase the counts when limits curve sharply or when the answer changes after recalculation.
What functions are supported?
You can use arithmetic, powers, pi, e, sin, cos, tan, sqrt, exp, log, abs, min, and max. Variables are r, theta, and z.
Can this calculate mass?
Yes. Enter density as the integrand. Keep the Jacobian option checked. The result estimates total mass over the cylindrical region.
Why does my result differ from an exact answer?
This calculator uses numerical integration. Small differences can occur from sampling. Increase even panel counts and compare repeated runs for convergence.