Calculator Inputs
Use x, y, and z.
Supported functions include sin, cos, sqrt,
exp, ln, pow, min, and max.
Formula Used
For two variables, the calculator evaluates ∫ from c to d ∫ from a(y) to b(y) f(x,y) dx dy.
For three variables, it evaluates ∫∫∫ f(x,y,z) d(inner) d(middle) d(outer).
The selected order defines inner, middle, and outer variables. Inner bounds may use outer variables.
h Σ f(a+(i+0.5)h)
h[(f(a)+f(b))/2 + Σf(xᵢ)]
h/3[f₀+fₙ+4Σfodd+2Σfeven]
The error estimate is the difference between a selected grid and a refined grid. The average value is the integral divided by region measure.
How to Use This Calculator
- Enter an integrand using
x,y, andz. - Select single, double, or triple integration.
- Choose the integration order from inner to outer.
- Enter bounds for active variables.
- Use outer variables inside inner bounds when needed.
- Select a method and interval count.
- Press calculate. Results appear above the form and below the header.
- Export the report as CSV or PDF.
Example Data Table
| Case | Function | Order | Bounds | Expected Use |
|---|---|---|---|---|
| Rectangular area | x*y |
dx then dy |
x: 0 to 2, y: 0 to 3 |
Basic double integral check. |
| Triangular region | x+y |
dx then dy |
x: 0 to y, y: 0 to 2 |
Variable upper bound test. |
| Solid volume | x*y*z |
dx then dy then dz |
0 to 1 for all variables |
Triple integral benchmark. |
| Radial style | x*z |
dx then dz |
Use x as radius. |
Add coordinate factors inside the integrand. |
Article: Multivariable Definite Integral Analysis
Purpose
A multivariable definite integral measures accumulation across a region. The region may be a rectangle, triangle, curved slice, box, or solid. This calculator focuses on practical numerical evaluation. It helps when an antiderivative is hard, long, or not available.
Flexible Bounds
Many real problems use dependent bounds. An inner limit can change as the outer variable changes. This feature is important in area, mass, probability, heat flow, and engineering models. The order field controls this structure. For example, dx then dy means x is integrated first. Its bounds may use y.
Numerical Methods
The midpoint rule samples the center of each subinterval. It is stable and simple. The trapezoidal rule connects endpoint values with straight lines. Simpson's rule uses parabolic arcs. It often gives strong accuracy for smooth functions. More intervals usually improve the estimate, but they also increase processing time.
Accuracy Review
The calculator performs a refinement check. It computes the integral with the selected grid. Then it repeats the process with a denser grid. The difference becomes an error estimate. This is not a formal proof. It is a useful warning signal. A large error suggests more intervals, smoother bounds, or a different method.
Graph Meaning
The graph shows the outer variable against a slice value. In a double integral, each point is the inner integral for one outer value. In a triple integral, each point represents a two-dimensional slice. This view helps reveal spikes, sign changes, and unstable regions.
Best Practice
Start with a simple known case. Confirm the order and bounds. Increase intervals slowly. Compare methods when results disagree. Put coordinate factors, such as a polar radius, directly in the integrand. Export the result when you need a record for reports, worksheets, or audit notes.
FAQs
1. What variables can I use?
You can use x, y, and z. The selected integration order decides which variables are active. Unused variables should not appear unless their value is supplied through the chosen bounds or order.
2. Can bounds depend on other variables?
Yes. Inner bounds may depend on variables integrated later. For dx then dy, x bounds may contain y. For dx then dy then dz, x may use y or z, and y may use z.
3. Which method should I choose?
Use Simpson for smooth functions. Use trapezoidal for steady endpoint behavior. Use midpoint when endpoint values are unstable or undefined. Compare methods if the result is sensitive.
4. Why are Simpson intervals changed?
Composite Simpson integration requires an even number of intervals. If you enter an odd count, the calculator increases it by one. This keeps the method valid.
5. What does region measure mean?
Region measure is the integral of 1 over the same bounds. In two dimensions it represents area. In three dimensions it represents volume.
6. Is the result exact?
No. The result is numerical. It approximates the definite integral using grid-based rules. Increase intervals and compare methods for stronger confidence.
7. How do I handle polar coordinates?
Use x as radius and y as angle if desired. Add the Jacobian factor inside the function, such as x*f(x,y), before calculating.
8. What do the export buttons include?
The CSV and PDF reports include the function, order, method, bounds, integral, error estimate, region measure, average value, and plotted sample data.