How to use (3 steps)
- Keep Auto unless you want a specific method; choose degrees/radians once.
- Start from the preset or edit f(x) and [a,b]; tolerance controls when adaptive methods stop.
- Scroll to Result for the value and a short meaning, then review the step log or copy LaTeX/URL.
Results
How it's calculated
Teacher notes
- Adaptive Simpson, Romberg, Gauss-Legendre (2/3/5 nodes), and Monte Carlo are available, with nodes and error estimates recorded in the step log.
- Infinite intervals expand exponentially while monitoring convergence, and a warning appears if divergence is suspected.
FAQ
Which integration methods are available?
Composite trapezoid, Simpson, adaptive Simpson, Romberg, Gauss-Legendre (2/3/5 nodes), and Monte Carlo are implemented. Each run records nodes, weights, and error estimates in the How it's calculated log.
How do you handle infinite limits or oscillatory integrals?
The integrator expands the interval exponentially while monitoring convergence, and raises an alert when divergence is suspected. Trigonometric functions automatically respect the degree/radian setting.
What is the probability mode for?
Probability mode is for integrating a density over an interval, such as finding the mass between two bounds. It is not a symbolic CDF table, so make sure your function really represents a density on the interval you are using.
Why can different methods disagree slightly?
Different methods sample the interval in different ways, so small differences are normal on difficult functions. If the interval is improper, highly curved, or oscillatory, compare the step log and tighten the tolerance before trusting the last digit.
When should I distrust the numeric answer?
Treat the estimate cautiously when the warning says the integral may diverge, when the method keeps refining without stabilising, or when the result meaning contradicts the graph. In those cases, review the interval and the function before using the number downstream.
How to interpret the output
Numeric estimate first, proof second
This tool is designed for fast numeric checking, not symbolic antiderivatives. Use it to decide whether a value is plausible, then move to a derivation page or notebook if you need a formal proof.
Use the graph and steps together
The shaded plot helps you confirm the interval and sign of the area, while the step log shows how the method sampled the region. If those two views tell different stories, the setup usually needs another look.
Probability interpretation
In probability mode, the result is the area under the density across the chosen interval. A result outside [0,1] is a sign that the function, bounds, or mode do not match a valid probability interpretation.
Improper-integral checklist
For ±inf bounds or endpoint singularities, read the warning text and compare two methods before reporting a final value. Agreement across methods is more informative than a single number on its own.
Suggested next steps
If you want to explain the setup to students, keep the result card visible and walk through the step log line by line. If you need a quick cross-check, copy the share URL and compare another method on the same interval.