Numerical Integration Calculator

Calculator

Input mode

Use data when measurements are discrete.

Method

Simpson needs an even interval count.

Units (optional)

Shown next to the integral value.

Function f(x)

Allowed: + − × ÷ ^, pi, e, sin, cos, exp, ln, sqrt.

Lower bound a

Upper bound b

Intervals n

Higher n usually improves accuracy.

y label (optional)

Data points (x,y)

Any spacing works. Example: 0,0 then 1,0.84.

i	x	sin(x)
0	0.000000	0.000000
1	0.523599	0.500000
2	1.047198	0.866025
3	1.570796	1.000000
4	2.094395	0.866025
5	2.617994	0.500000
6	3.141593	0.000000

Formula used

Trapezoidal rule

I ≈ (h/2)[f(x₀) + 2Σ f(xᵢ) + f(xₙ)]

Reliable for measured data and smooth curves.

Simpson's 1/3 rule

I ≈ (h/3)[f(x₀)+4Σ f(xᵢ odd)+2Σ f(xᵢ even)+f(xₙ)]

Often very accurate for smooth functions.

Midpoint rule

I ≈ h Σ f(a+(i+1/2)h)

Good when endpoints are noisy or unstable.

Here, h = (b − a)/n, and xᵢ = a + i·h. The “error hint” compares n and 2n to show convergence.

How to use

Pick Function of x or Tabulated data.
For a function, enter f(x), bounds a, b, and intervals n.
Choose one method or compare all methods.
Press Calculate. Results appear above the form.
Use Download CSV or Download PDF for reporting.

Tip: if Simpson’s rule fails, make n even.

Integration in field and motion models

Numerical integration converts sampled forces, fields, and spectra into physically meaningful totals. For example, impulse equals ∫F(t)dt, charge equals ∫I(t)dt, and displacement equals ∫v(t)dt. With experimental sensors, values arrive as discrete points, so a stable rule is needed to approximate the area under the curve.

Choosing step size with measurable targets

The interval count n controls the step h=(b−a)/n. If the signal varies rapidly, smaller h captures curvature and reduces bias. A practical target is convergence: double n and watch the estimate change. When |I(2n)−I(n)| becomes small compared with your measurement uncertainty, refinement beyond that point adds little value. For many labs, keeping the relative change below 0.5% is a sensible threshold, especially when calibration drift or digitization noise dominates the uncertainty budget.

Trapezoidal rule for sampled measurements

The trapezoidal rule assumes the curve is linear between samples. For unequal spacing, it sums (xᵢ₊₁−xᵢ)(yᵢ+yᵢ₊₁)/2. It performs well for oscilloscope traces, calorimetry logs, and tabulated cross‑section data, and it remains robust when noise makes higher‑order curvature fitting unreliable.

Simpson’s rule for smooth analytical curves

Simpson’s 1/3 rule blends parabolic arcs across pairs of sub‑intervals, using endpoint and midpoint weighting. It typically achieves higher accuracy than trapezoids for smooth functions, such as Gaussian beam profiles or potential energy curves. Because it requires an even n, the calculator flags odd interval counts to prevent misuse.

Midpoint rule to reduce endpoint sensitivity

The midpoint rule samples each sub‑interval at its center. This can be advantageous when endpoints have discontinuities or when boundary values are uncertain. In physics, it can reduce sensitivity to singular behavior near a, such as 1/√x, by avoiding direct evaluation at the boundary while still representing the interior contribution.

Reporting results for reproducible analysis

Professional reporting should include the chosen method, bounds, n, and a trace of sampled points. Exporting a CSV supports lab notebooks and code review, while a PDF summary works for quick sharing. Comparing multiple methods helps detect instability: large divergence suggests insufficient resolution, sharp discontinuities, or an input expression that becomes non‑finite inside the range.

FAQs

Which method should I use for experimental sensor data?

Use the data mode with the trapezoid sum. It matches how instruments sample signals and stays stable under noise. Increase point density where the signal bends sharply, and confirm convergence by refining the sampling rate.

Why does Simpson’s rule require an even number of intervals?

Simpson’s 1/3 rule fits parabolas across pairs of sub‑intervals. Each parabola spans two intervals, so the total count must be even. If n is odd, increase it by one to satisfy the pairing.

What does the error hint actually mean?

The calculator recomputes the primary method using n and 2n intervals and reports |I(2n)−I(n)|. A smaller value indicates the estimate is stabilizing. It is a heuristic, not a guaranteed bound for every function.

Can I integrate functions with singularities or discontinuities?

You can, but accuracy may degrade. Split the range into safer segments around the problematic point, avoid evaluating exactly at the singularity, and use higher resolution near sharp changes. Compare methods to detect instability.

How do I format tabulated data inputs?

Enter one point per line as x,y or x y. The tool sorts points by x and ignores duplicates. Provide at least two points. Units are optional and only annotate the reported integral.

What should I include when exporting results to a report?

Include f(x) or the data source, bounds, method, n, and the convergence difference. Attach the exported sample table so others can reproduce the estimate. If results differ across methods, note resolution limits or discontinuities.