Romberg Integration Calculator

Enter a function f(x), set your lower limit (a) and upper limit (b), and choose the number of iterations — the Romberg Integration Calculator computes the definite integral using Richardson extrapolation for high accuracy. You get the integral result, the Romberg table of successive approximations, and a convergence breakdown across iterations.

Use standard math notation: sin(x), cos(x), exp(x), log(x), sqrt(x), x^2, etc. Use * for multiplication.

The lower bound of integration.

The upper bound of integration.

More iterations give greater accuracy. Values between 3 and 7 are typical.

Results

Definite Integral Result

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Iterations Completed

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Initial Trapezoidal Estimate R(1,1)

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Final Richardson Extrapolation R(n,n)

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Improvement Over Trapezoidal (%)

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Romberg Approximation Convergence

Results Table

Frequently Asked Questions

What is Romberg integration?

Romberg integration is a numerical method for approximating definite integrals. It applies Richardson extrapolation repeatedly to a sequence of trapezoidal rule estimates, progressively eliminating error terms to achieve a highly accurate result without needing a very large number of sub-intervals.

How does Richardson extrapolation improve the trapezoidal rule?

Richardson extrapolation combines two estimates of an integral computed with different step sizes to cancel out leading error terms. The formula R(k, m) = R(k, m-1) + [R(k, m-1) - R(k-1, m-1)] / (4^(m-1) - 1) is applied iteratively, each column of the Romberg table being more accurate than the last.

How many iterations should I use?

For most smooth functions, 4 to 6 iterations provide excellent accuracy. Using more iterations generally increases precision but with diminishing returns. For highly oscillatory or discontinuous functions, consider using specialized methods or subdividing the interval manually.

What functions can I enter in the calculator?

You can enter any standard mathematical function of x, such as polynomials (x^2 + 3*x), trigonometric functions (sin(x), cos(x), tan(x)), exponentials (exp(x)), logarithms (log(x)), and square roots (sqrt(x)). Always use * for multiplication, e.g. 2*x rather than 2x.

Why might the result differ from an exact analytical integral?

Romberg integration is a numerical approximation. While it is very accurate for smooth functions, floating-point arithmetic and the finite number of iterations mean results are not always exact. Increasing the number of iterations reduces this error, and for polynomials of moderate degree the result is typically exact to machine precision.

What does the Romberg table show?

The Romberg table is a triangular array where each row k starts with a trapezoidal estimate using 2^(k-1) sub-intervals. Moving across the columns (increasing m) applies Richardson extrapolation, with R(n, n) in the bottom-right corner being the most accurate estimate.

Is Romberg integration better than Simpson's rule?

Romberg integration is generally more accurate than Simpson's rule for the same number of function evaluations, because it achieves higher-order accuracy through repeated extrapolation. Simpson's rule corresponds roughly to the second column (m=2) of the Romberg table, while further columns improve accuracy significantly.

What happens if the upper and lower limits are the same?

If the lower limit equals the upper limit, the definite integral is zero by definition, since there is no interval over which to integrate. The calculator will return 0 in this case.

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