Approximate a definite integral with weighted quadratic panels. The calculation trail makes the reported simpson estimate easier to reproduce.
Composite Simpson’s rule weights endpoints 1, odd interior samples 4, and even interior samples 2.
For this page, simpson’s rule is interpreted under the stated convention and input order.
It provides high accuracy for smooth functions and is exact for polynomials through degree three.
The subinterval count must be even. Discontinuities and singularities still need interval splitting or specialized methods. If the Simpson’s Rule assumptions do not fit, consider automatic Simpson integration.
Compare the result with the worked example's scale before relying on the reported simpson estimate.
For x⁴ on 0 to 2, the estimate approaches 32/5=6.4. This Simpson’s Rule example can be compared with trapezoid comparison.
Divide evenly, apply alternating 4 and 2 weights, add endpoints, and multiply by h/3.
Source values may arrive in a different order from the form. Map them explicitly to Function f(x), Lower bound, Upper bound and Even subinterval count, normalize units, and retain enough precision for the next step after simpson estimate.
Increasing even n normally reduces smooth-function error quickly. Watching this response separates a data-entry mistake from an unexpected but valid value.
If Function f(x), Lower bound, Upper bound and Even subinterval count are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Trapezoidal rule accepts any n but usually converges more slowly on smooth curves. This page deliberately reports only the former interpretation.
Label the output as simpson estimate in notes or tables. Store Function f(x), Lower bound, Upper bound and Even subinterval count beside it when the result will be reused in a later stage.
The working data for Simpson’s Rule include Function f(x), Lower bound, Upper bound, and Even subinterval count. Check nearby points rather than only the displayed answer. Smooth neighboring behavior supports derivative and quadrature assumptions, while abrupt changes suggest a corner, pole, or unresolved feature that deserves a separate interval.
For this simpson’s rule result, state the numerical method when it affects interpretation. A finite-difference derivative, sampled limit, and panel-based integral can agree closely with an exact value while carrying different sources of uncertainty.
Quadratic panels span pairs of subintervals.
Through cubic.
Not without splitting.