Romberg Integration Calculator

How it is calculated

You choose one of four built-in functions (x², sin x, eˣ, 1/(1+x²)), the limits a and b (b must be greater than a) and a number of levels L from 1 to 10. The first column of the Romberg table holds trapezoid estimates with 1, 2, 4, 8 … intervals: R(i,0) = ½·R(i−1,0) + h·Σ f(midpoints), with h halved each row. Each further column applies Richardson extrapolation: R(i,j) = (4j·R(i,j−1) − R(i−1,j−1)) ÷ (4j − 1).

The answer shown is the bottom-right entry of the table, rounded to six decimals. Only that final estimate is displayed, not the whole table. L = 1 is just one trapezoid over the whole interval. A decimal number of levels is cut down to a whole number, and values outside 1–10 give no result. The sine function works in radians.

Common mistakes

• Entering limits in degrees for sin x. The function uses radians, so the integral from 0 to 180 is not the integral over half a turn; use 0 to 3.14159 instead.
• Using too few levels on a wide or oscillating interval. For sin x over 0 to 100 with 3 levels the estimate is far from the true value, so raise the levels and check that the answer settles.
• Choosing 1 level and expecting accuracy. One level is a single trapezoid; for sin x from 0 to π it returns 0.000000 instead of 2.
• Entering the limits in the wrong order (a larger than b). The tool needs b greater than a and gives no result otherwise; swap them and change the sign yourself.
• Expecting to type any formula. Only the four listed functions are available, and levels above 10 are rejected.