Calculator Inputs
Example Data Table
| Example | Mode | Integrand | Bounds | Intervals | Expected Insight |
|---|---|---|---|---|---|
| Unit square energy | Double Cartesian | x^2 + y^2 | x: 0 to 1, y: 0 to 1 | 10 by 10 | Checks smooth polynomial integration. |
| Oscillating field | Double Cartesian | sin(x)*cos(y) | x: 0 to 3.14159, y: 0 to 1.5708 | 20 by 20 | Shows sign changes and cancellation. |
| Unit cylinder slice | Double Polar | r^2 | r: 0 to 1, θ: 0 to 6.28318 | 20 by 20 | Uses Jacobian weighting through r. |
| Mass density block | Triple Cartesian | x+y+z | 0 to 1 on all axes | 8 by 8 by 8 | Approximates total volume density. |
Formula Used
This calculator estimates multivariable integrals numerically when a closed-form antiderivative is inconvenient or unavailable.
For a rectangular double integral, the target quantity is
∬ f(x,y) dA,
approximated over a grid with cell widths
Δx = (xmax - xmin) / nx
and
Δy = (ymax - ymin) / ny.
Midpoint rule uses center samples:
Integral ≈ Σ f(xi*, yi*) Δx Δy.
Trapezoidal rule uses corner-edge weighting:
Integral ≈ Σ wx wy f(xi, yi) Δx Δy.
Simpson rule uses alternating weights 1, 4, and 2:
Integral ≈ (Δx Δy / 9) Σ wx wy f(xi, yi).
Triple integrals extend this pattern with
Δz.
Simpson scaling becomes
Δx Δy Δz / 27.
For polar coordinates, the Jacobian factor is required:
∫∫ f(r,θ) r dr dθ.
The extra
r
term converts polar area to Cartesian area correctly.
How to Use This Calculator
- Select the integration mode matching your domain shape and dimension.
- Enter an integrand using supported variables and standard function names.
- Provide lower and upper bounds for each active coordinate.
- Choose interval counts. Use even counts for Simpson calculations.
- Click Calculate Integral to compute the approximation.
- Review the result, graph, and sample contribution table.
- Export the displayed report using CSV or PDF buttons.
Frequently Asked Questions
1. What does this calculator solve?
It approximates double and triple integrals over rectangular grids and double integrals in polar form. It helps estimate area, volume, mass, energy, and accumulated quantities from user-defined functions.
2. When should I use midpoint, trapezoidal, or Simpson?
Midpoint is simple and stable. Trapezoidal is useful for boundary-aware averaging. Simpson is usually more accurate for smooth functions, but it requires even interval counts along every active axis.
3. Why does the polar mode multiply by r?
Polar area elements are not plain rectangles. The Jacobian factor r adjusts each small sector correctly, so the integral matches the actual Cartesian area covered by the polar grid.
4. Which variables can I enter?
Use x and y for double Cartesian mode, x, y, and z for triple Cartesian mode, and r with theta for polar mode. Constants pi and e are also supported.
5. Can this replace symbolic integration software?
No. This page is a numerical approximation tool. It estimates integrals from sampled grid values. Symbolic software is still better when you need exact algebraic antiderivatives.
6. Why does Simpson sometimes show an error?
Simpson weighting alternates across evenly spaced sample points. That pattern only works correctly when each active axis uses an even number of intervals, such as 10, 20, or 40.
7. What does the graph represent?
The plot visualizes sampled values over the active grid. For triple mode, it shows a mid-volume z-slice. For polar mode, it displays the weighted contribution grid f(r,θ)·r.
8. How can I improve accuracy?
Increase interval counts, choose Simpson for smooth functions, verify bounds carefully, and compare repeated runs. Finer grids usually improve estimates, though computation time also increases.