Calculator inputs
Example data table
| Case | Integrand | Bounds | Method | Partitions | Approximate integral |
|---|---|---|---|---|---|
| Double integral | x + y | x: [0,1], y: [0,2] | Simpson | 12 × 12 | 3.000000 |
| Double integral | x * y | x: [0,2], y: [0,3] | Trapezoidal | 18 × 18 | 9.000000 |
| Triple integral | x**2 + y**2 + z | x: [0,1], y: [0,1], z: [0,1] | Simpson | 12 × 12 × 12 | 1.166667 |
| Double integral | sin(x) + cos(y) | x: [0,π], y: [0,π/2] | Midpoint | 20 × 20 | 6.283185 |
Formula used
Double integral over a rectangular region: For a function f(x,y) on x ∈ [a,b] and y ∈ [c,d], the target quantity is ∫∫R f(x,y) dA.
Triple integral over a box region: For a function f(x,y,z) on x ∈ [a,b], y ∈ [c,d], and z ∈ [e,f], the target quantity is ∫∫∫V f(x,y,z) dV.
Midpoint rule: The calculator samples each subregion at its center and multiplies the sum by the subregion measure.
Trapezoidal rule: The calculator evaluates corner grid points and applies edge weights through a product trapezoidal scheme.
Simpson rule: The calculator applies product Simpson weights. In two variables, the factor is ΔxΔy/9. In three variables, the factor is ΔxΔyΔz/27.
Average value over the region: Average value = integral ÷ region measure. This normalizes the result by area for 2D or volume for 3D.
How to use this calculator
- Select double or triple integration.
- Enter the integrand using x, y, and z when needed.
- Set lower and upper bounds for each active variable.
- Choose a numerical method and partition counts.
- Adjust display decimals and plot density if desired.
- Press Calculate integral to view the result above the form.
- Review the refined estimate, stability note, and convergence graph.
- Download CSV or PDF for documentation or sharing.
FAQs
1. What types of functions can I enter?
You can enter algebraic and many standard transcendental expressions using x, y, and z. Supported functions include sin, cos, tan, sqrt, abs, log, ln, exp, pow, min, max, floor, ceil, and round.
2. Why does Simpson sometimes change my partition count?
Simpson’s rule needs an even number of subintervals along every active axis. If you enter an odd value, the calculator automatically increases it and reports that adjustment in the notes.
3. What is the difference between the main and refined estimates?
The main estimate uses your current partition counts. The refined estimate doubles the grid density, then recalculates the integral. Comparing them helps you judge numerical stability and whether the mesh is fine enough.
4. When should I use midpoint, trapezoidal, or Simpson?
Midpoint is simple and often solid for smooth functions. Trapezoidal is useful for grid-based intuition. Simpson usually gives better accuracy for smooth curves and surfaces, especially when enough even partitions are used.
5. What does the average value tell me?
The average value divides the computed integral by the region’s area or volume. It represents the mean height of the function over the chosen domain, which is often easier to interpret than the raw integral.
6. Why is the triple-integral graph shown as a slice?
A full three-variable function cannot be displayed directly on one surface plot. The calculator therefore shows an x-y slice at the midpoint of z, which gives a practical visual snapshot of the 3D integrand.
7. What should I do if stability is weak?
Increase partitions along the variables where the function changes quickly. Sharp peaks, oscillations, and steep gradients usually need denser grids. Switching to Simpson for smooth functions can also improve the estimate.
8. Can this calculator handle discontinuities or singularities?
It can approximate some difficult cases, but accuracy may degrade near discontinuities or singular points. If the function becomes undefined inside the bounds, the calculator stops and asks you to revise the formula or region.