Calculator Inputs
Enter a function in terms of x and y.
Use explicit multiplication like x*y and powers like x^2.
Example Data Table
| Function | x lower | x upper | y lower | y upper | Method | x steps | y steps | Expected value |
|---|---|---|---|---|---|---|---|---|
| x*y + x^2 | 0 | 2 | 1 | 3 | Simpson | 20 | 20 | 13.33333333 |
| sin(x) + y^2 | 0 | 1 | 0 | 2 | Midpoint | 40 | 40 | Approximate numerically |
Formula Used
If f(x, y) is integrable on R = [a, b] × [c, d], then
∬R f(x, y) dA = ∫ab [ ∫cd f(x, y) dy ] dx = ∫cd [ ∫ab f(x, y) dx ] dy
Average value = Double integral / Region area
For one dimension,
∫ f(t) dt ≈ h Σ f(ti + h/2)
∫ f(t) dt ≈ h [ (f(t0) + f(tn))/2 + Σ f(ti) ]
∫ f(t) dt ≈ h/3 [ f(t0) + f(tn) + 4Σ f(odd points) + 2Σ f(even points) ]
How to Use This Calculator
- Enter a valid function using variables x and y.
- Provide rectangular bounds for both variables.
- Choose a numerical method for repeated integration.
- Set subdivisions for x and y. Higher values usually improve accuracy.
- Choose display decimals and a tolerance for comparing both orders.
- Press Compute Double Integral.
- Review the two iterated integrals, their difference, and average value.
- Export the shown result as CSV or PDF when needed.
FAQs
1. What does this calculator verify?
It numerically compares the two repeated integration orders on a rectangular region. Close values support the expected theorem behavior, but they do not replace a formal proof of continuity or absolute integrability.
2. Can it handle symbolic integration exactly?
No. This version performs numerical approximation. It is designed to compare orders, estimate the integral, and build intuition when an exact antiderivative is difficult or unnecessary.
3. Which functions can I enter?
You can use x, y, constants, parentheses, powers, and common functions such as sin, cos, tan, sqrt, abs, exp, log, log10, min, max, and pow.
4. Why do both orders sometimes differ slightly?
The theorem gives equality under its assumptions, but numerical rules introduce approximation error. Increasing subdivisions or switching to Simpson rule often reduces the observed mismatch.
5. Why are my Simpson steps adjusted automatically?
Simpson rule requires an even number of subintervals. If you enter an odd value, the calculator increases it by one so the formula remains valid.
6. Does this work for non-rectangular regions?
This page is built for rectangular regions only. For triangular or curved regions, you would need variable bounds or a coordinate transformation.
7. What is the average value result?
It is the double integral divided by the region area. This tells you the mean height of the surface over the selected rectangle.
8. How large should the subdivisions be?
Start with 20 to 40 per variable. Increase them for oscillating or sharply changing functions. Larger values improve accuracy but require more computation.