Enter a polynomial, bounds, and an even number of subintervals, and instantly get the Simpson’s 1/3 rule estimate — compared against the exact integral.
Simpson’s 1/3 Rule Calculator
How It’s Calculated
Simpson’s 1/3 rule fits a parabola through each pair of strips instead of a straight line, which is why it needs an even number of subintervals: ∫ f(x) dx ≈ (h/3)·[f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(b)] — odd-indexed points get weight 4, even-indexed interior points get weight 2. Because it captures curvature rather than just a straight edge, Simpson’s rule is exact for any polynomial up to degree 3, and noticeably more accurate than the trapezoidal rule for the same n on curved functions.
Worked Example
For f(x) = x² on [0,4] with n = 4: h = 1, points 0,1,2,3,4 with f-values 0,1,4,9,16. Estimate = (1/3)·[0 + 4(1) + 2(4) + 4(9) + 16] = (1/3)·[0+4+8+36+16] = 64/3 ≈ 21.33 — exactly matching the true integral, since x² is degree 2 and Simpson’s rule is exact through degree 3.
Frequently Asked Questions
Q: Why does n have to be even?
Simpson’s rule works by grouping subintervals in pairs, fitting one parabola across each pair — an odd number of subintervals would leave one strip without a partner.
Q: Is Simpson’s rule always more accurate than the trapezoidal rule?
For the same n, yes, in general — fitting parabolas captures curvature that straight-line trapezoids miss, which is why Simpson’s rule is exact for polynomials up to degree 3 while the trapezoidal rule is only exact for straight lines.
Simpson’s rule is a standard Numerical Methods topic across VTU Engineering Mathematics. Compare with the Trapezoidal Rule →