Three error curves race downward as the partition doublesLive current inputs
Math & Statistics
Integral Approximation Calculator
Approximate the integral of a sine wave plus a constant offset, compare three composite rules over a doubling sequence, and measure observed error convergence against the exact antiderivative.
Exact integral -
Finest Simpson result -
Finest trapezoid result -
Finest midpoint result -
Finest Simpson error -
Trapezoid observed order -
Simpson observed order -
Finest node count -
NUMERICAL-INTEGRATION CONVERGENCE
Three error curves race downward as the partition doubles
A log-error chart separates midpoint, trapezoid, and Simpson behavior, while a compact wave inset shows the actual oscillation being integrated.
DOUBLING STUDY
Approximation, absolute error, and error ratio at every resolution
Each row doubles n, making empirical order visible without changing the function or interval.
| Subintervals n | Midpoint error | Trapezoid error | Simpson error | Trap error ratio | Simpson error ratio |
|---|
CONVERGENCE SETUP
Resolve the wave rather than merely increasing n
- Enter amplitude, frequency, and offset.
- Choose bounds covering the intended oscillation window.
- Start with an even n so all three rules are comparable.
- Use several doubling levels to reveal a trend.
- Inspect error order before trusting the smallest displayed error.
RESOLUTION VERSUS METHOD
A high-order rule cannot rescue a partition that misses the oscillation
Simpson's formal fourth-order behavior appears only when the function is smooth and the grid resolves its changes. Coarse samples can alias a sine wave.
Midpoint and trapezoid are both second-order for smooth functions but usually carry opposite curvature biases over short sections. Their errors need not decline monotonically across every oscillatory interval.
COMPOSITE APPROXIMATION
Compare methods on one exact benchmark and a controlled resolution sequence
The exact sine-plus-offset antiderivative supplies a transparent benchmark. Error ratios show whether each method approaches its expected asymptotic order.
Detailed calculation process and general formulas
f(x)=A sin(kx)+CI_exact=A[cos(kl)-cos(ku)]/k+C(u-l)E_method(n)=|I_method(n)-I_exact|ratio=E(n)/E(2n)order=log2(ratio)Symbols, meanings, and units
- A
- sine amplitudef-unit
- k
- angular frequencyradian/x-unit
- C
- constant offsetf-unit
- n
- composite subinterval countcount
- E
- absolute integration errorf-unit*x-unit
- order
- empirical convergence orderdimensionless
CONVERGENCE DIAGNOSTICS
Use order and error together
A tiny error can be accidental cancellation; a stable order provides stronger evidence.
Finest accuracy
-Absolute error measures the current endpoint result.
Method regime
-Observed order indicates whether the sequence behaves asymptotically.
Evaluation cost
-Node count shows the resolution paid for the final result.
Decision takeaway: Refine until the error trajectory is stable, not merely until one rounded total looks acceptable.
NUMERICAL STUDY NOTES
Document these details with an approximation result
- Function and parameter units
- Bounds and orientation
- Rule and node convention
- Starting n and number of refinements
- Exact or higher-accuracy benchmark
- Observed error order
Applied decisions
When convergence history is more valuable than one approximation
Oscillatory load
A sinusoidal rate is integrated across several partial cycles.
What the result clarifies: The wave inset exposes whether the starting grid can see the oscillation.
Algorithm regression test
A numerical library update changes endpoint weights.
What the result clarifies: The error-order sequence reveals degraded behavior even when one total remains close.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
Observed convergence can be disrupted by aliasing, cancellation, floating-point limits, discontinuities, or insufficient resolution. The page benchmarks the entered smooth sine model only.
Integral Approximation Calculator | Convergence of Midpoint, Trapezoid, and Simpson Rules FAQ
Why use log error?
Orders of magnitude and convergence slopes are easier to compare on a logarithmic scale.
Why can an error be exactly zero?
Symmetry or cancellation can make a particular partition exact; it does not guarantee the method is exact generally.
What ratios are expected?
Roughly four for second-order midpoint and trapezoid rules and sixteen for fourth-order Simpson behavior.
Does more n always help?
Usually until rounding or model-evaluation noise becomes significant.