Adaptive Quadrature Calculator

Calculator Inputs

Function f(x)

Lower Bound

Upper Bound

Tolerance

Maximum Depth

Minimum Interval Width

Plot Density Points

Reference Simpson Panels

Supported syntax: x, pi, e, +, -, *, /, ^, parentheses, sin, cos, tan, asin, acos, atan, sqrt, exp, log, log10, ln, abs, pow, min, max, sec, csc, cot, sinh, cosh, tanh.

Example Data Table

Function	Interval	Tolerance	Approximate Integral	Why It Is Useful
x^4 - 2*x + 1	[0, 2]	1e-8	4.4000000000	Checks a smooth polynomial with an exact benchmark.
sin(x^2)	[0, 3]	1e-6	0.7735625269	Shows adaptive refinement on an oscillatory integrand.
exp(-x^2)	[-2, 2]	1e-6	1.7641627815	Useful for bell-shaped functions and fast decay.

Formula Used

This calculator uses adaptive Simpson quadrature. It compares one Simpson estimate on an interval with the sum of two Simpson estimates on the split halves.

Single-interval Simpson estimate:
S(a,b) = (b - a) / 6 × [f(a) + 4f((a+b)/2) + f(b)]

Recursive split:
[a,b] → [a,m] and [m,b], where m = (a+b)/2

Local error estimate:
Error ≈ |S(a,m) + S(m,b) - S(a,b)| / 15

Accepted corrected estimate:
Integral ≈ S(a,m) + S(m,b) + [S(a,m) + S(m,b) - S(a,b)] / 15

Smaller tolerances usually improve accuracy, but may increase interval count, recursion depth, and sampled function evaluations.

How to Use This Calculator

Enter a valid function of x, such as sin(x^2) or exp(-x^2).
Set the lower and upper bounds for the definite integral.
Choose a tolerance. Smaller tolerances request tighter local control.
Set a maximum recursion depth to limit repeated interval splitting.
Set a minimum interval width to prevent excessive refinement on tiny panels.
Choose plot density and reference Simpson panels for diagnostics.
Press Compute Integral to see the result, graph, interval table, and convergence diagnostics above the form.
Use the CSV and PDF buttons to export the accepted interval data and summary.

FAQs

What does adaptive quadrature do?

It estimates a definite integral by splitting the interval where the function is harder to approximate. Smooth regions get fewer panels, while rapidly changing regions get more refinement.

Why use adaptive Simpson refinement here?

Adaptive Simpson’s rule is accurate for many smooth functions and gives a practical local error estimate. That lets the calculator refine only where needed and stop efficiently.

What does the tolerance input mean?

Tolerance is the target local error scale for recursive refinement. Smaller values usually improve accuracy, but they also increase subdivisions, runtime, and sampled points.

Why can depth limits matter?

Very oscillatory, sharp, or nearly singular functions may keep demanding smaller intervals. The depth limit prevents endless refinement and reports when the cap influenced the result.

Why is there a reference estimate too?

The reference estimate uses a dense composite Simpson pass for comparison. It is a diagnostic check, not the main adaptive answer, and it can help reveal suspicious differences.

Which functions can I type?

You can use x, numbers, parentheses, and functions like sin, cos, tan, sqrt, exp, log, log10, ln, abs, sinh, cosh, tanh, sec, csc, cot, pow, min, and max.

Why do some functions fail on the interval?

If the function becomes undefined, infinite, or non-real at sampled points, numerical integration cannot continue safely. Examples include log(x) at x ≤ 0 or division by zero.

What do the graph markers show?

The line shows the function over the interval, while markers show adaptive sample locations gathered during recursion. Dense clusters indicate areas that required more numerical attention.