Romberg Integration Calculator

Enter f(x), the interval a to b, and a refinement level k to compute a high-accuracy definite integral using Romberg integration: successive trapezoidal refinement with Richardson extrapolation.

Input

Enter the integrand f(x), the interval, and a refinement level to compute the definite integral with Romberg integration: trapezoidal refinement plus Richardson extrapolation.

Integrand f(x)

The variable is x. Arithmetic, power (^), functions like sin, cos, exp, log, sqrt, and constants pi, e are supported.

Lower bound a

Upper bound b

Refinement level k

From 1 to 20. Higher levels give higher accuracy.

Result

Approximate integral R(k,k)

0.7853981634

Interval from a = 0 to b = 1

Refinement level k

Evaluation count

Estimated error

1.213030e-11

Romberg extrapolation table

Shows the trapezoidal rule R(i,0) for each level and the fully extrapolated value R(i,i) of that row.

Level i	Intervals	Trapezoid R(i,0)	Extrapolated R(i,i)
0	1	0.75	0.75
1	2	0.775	0.78333333
2	4	0.78279412	0.78552941
3	8	0.78474712	0.78539645
4	16	0.7852354	0.78539817
5	32	0.78535747	0.78539816
6	64	0.78538799	0.78539816

How it works

Romberg integration builds a sequence of composite trapezoidal approximations on intervals doubled by powers of two, then repeatedly applies Richardson extrapolation to reach higher-order accuracy.
Row i starts from the trapezoidal rule R(i,0) with 2^i intervals and extrapolates via R(i,j) = R(i,j-1) + (R(i,j-1) - R(i-1,j-1)) / (4^j - 1) to obtain R(i,i).
The headline result is R(k,k) at the maximum level k. For sufficiently smooth integrands, raising the level converges rapidly toward the true value.
The estimated error is the absolute difference between R(k,k) and R(k-1,k-1). It is a guideline rather than the exact error.
f(x) supports x with arithmetic, power (^), and elementary functions such as sin, cos, tan, exp, log, sqrt, abs, plus the constants pi and e.
If the interval contains singularities (such as a zero denominator) or divergence, an evaluation may become non-finite and the integral cannot be computed.

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