Simpson's Rule Calculator

Simpson’s rule approximates a definite integral by fitting parabolas through each pair of subintervals rather than rectangles. It is far more accurate than simple Riemann sums for smooth functions.

Simpson’s 1/3 rule (n must be even): integral ≈ (dx/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + … + 4f(x_{n-1}) + f(x_n)]

The alternating 4-2-4-2 pattern of coefficients is the signature of this rule. Each group of three points (two subintervals) fits one parabola. Requiring n to be even ensures the groups pair up correctly.

Simpson’s 3/8 rule (n must be divisible by 3): integral ≈ (3dx/8) * [f(x0) + 3f(x1) + 3f(x2) + 2f(x3) + 3*f(x4) + … + f(x_n)]

For a polynomial of degree 3 or less, both Simpson rules give the exact answer regardless of n. For a quadratic f(x) = ax^2 + bx + c, even n=2 (three points) is exact.

That last fact has a practical consequence for using this page. If you enter only a quadratic, every error it reports will be zero, which demonstrates nothing about the method beyond the fact that it is exact there. That is why the calculator takes a fourth-degree polynomial: degree 4 is the lowest degree at which Simpson’s rule makes an error at all. Put a value in the x⁴ box and the numbers become interesting; leave it empty and you get a row of zeroes and a trapezoidal comparison to show what the zeroes are worth.

The error in Simpson’s 1/3 rule is proportional to dx^4 (proportional to 1/n^4). That is dramatically better than the trapezoidal rule (1/n^2), since halving the step size reduces the error by a factor of 16 rather than 4. You can watch that happen on this page: integrate x⁴ over [0, 3], then double n and compare.

Gauss-Legendre quadrature and adaptive quadrature can do even better for smooth functions, but Simpson’s rule is what most engineering and physics courses use as the go-to numerical integration method.