Mathematics · Calculus
Simpson's Rule Calculator
Approximate an integral from endpoint and midpoint function values.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Weighted sum=0+4×1+4=8.
- Estimate=(2−0)×8÷6=2.6666666666666665.
Understand Simpson's rule
One idea, three depths
Choose how deeply to explain Simpson's rule
Simpson's rule: Approximate an integral from endpoint and midpoint function values.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Simpson's rule to answer this question: approximate an integral from endpoint and midpoint function values? Enter Start a, End b, f(a), and 2 other inputs; the calculator shows Simpson estimate. For example: For x² on 0 to 2, values 0,1,4 give the exact integral 8/3. The answer tells you Simpson estimate.
Age 15Explain it to a 15-year-oldConnect it to the formula
A quadratic through three equally spaced samples integrates to Simpson's weighted formula. The rule is area≈(b−a)[f(a)+4f(mid)+f(b)]/6. Its input values are Start a, End b, f(a), f(mid), f(b), and the main result is Simpson estimate. For example: For x² on 0 to 2, values 0,1,4 give the exact integral 8/3.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated simpson's rule relation over the valid real-number domain stated below. The implemented relation is area≈(b−a)[f(a)+4f(mid)+f(b)]/6, evaluated from Start a, End b, f(a), f(mid), f(b) to produce Simpson estimate. A quadratic through three equally spaced samples integrates to Simpson's weighted formula. The middle value must be evaluated at the interval midpoint.
Inputs and valid domain
- Start a must be a finite real number.
- End b must be a finite real number.
- f(a) must be a finite real number.
- f(mid) must be a finite real number.
- f(b) must be a finite real number.
Important boundary: The middle value must be evaluated at the interval midpoint.
The formula
area≈(b−a)[f(a)+4f(mid)+f(b)]/6
How the calculator works through it
It substitutes Start a, End b, f(a), f(mid), f(b) into the formula and exposes every numerical step above. The main output is Simpson estimate, accompanied by Interval midpoint, Weighted height sum.
Read the result correctly
The Simpson estimate is the direct answer to “approximate an integral from endpoint and midpoint function values.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For x² on 0 to 2, values 0,1,4 give the exact integral 8/3.
Where this model stops being reliable
The middle value must be evaluated at the interval midpoint.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Simpson's rule works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Simpson's rule uses area≈(b−a)[f(a)+4f(mid)+f(b)]/6. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives as rates of change
Rates of change explain the local behaviour captured or approximated by Simpson's rule.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Simpson's rule to accumulated change, area and continuous totals.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Start a, End b, f(a), f(mid), f(b).
- Evaluate the principal relationship: area≈(b−a)[f(a)+4f(mid)+f(b)]/6.
- Return Simpson estimate and check the domain conditions described above.
Python
from math import *
def simpson_rule_single(x1, x2, a, b, c) -> float:
return (((x2 - x1) * ((a + (4.0 * b)) + c)) / 6.0)
assert abs(simpson_rule_single(0, 2, 0, 1, 4) - 2.6666666666666665) < 1e-6 * max(1.0, abs(2.6666666666666665))
C
#include <assert.h>
#include <math.h>
double simpson_rule_single(double x1, double x2, double a, double b, double c) {
return (((x2 - x1) * ((a + (4.0 * b)) + c)) / 6.0);
}
int main(void) {
const double expected = 2.6666666666666665;
const double actual = simpson_rule_single(0, 2, 0, 1, 4);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double simpson_rule_single(double x1, double x2, double a, double b, double c) {
return (((x2 - x1) * ((a + (4.0 * b)) + c)) / 6.0);
}
int main() {
constexpr double expected = 2.6666666666666665;
const double actual = simpson_rule_single(0, 2, 0, 1, 4);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double simpson_rule_single(double x1, double x2, double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global simpson_rule_single
section .text
simpson_rule_single:
push rbp
mov rbp, rsp
sub rsp, 112
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd [rbp-40], xmm4
movsd xmm0, [rbp-16]
subsd xmm0, [rbp-8]
movsd [rbp-64], xmm0
mov rax, 0x4010000000000000
movq xmm0, rax
movsd [rbp-96], xmm0
movsd xmm0, [rbp-96]
mulsd xmm0, [rbp-32]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-24]
addsd xmm0, [rbp-88]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-80]
addsd xmm0, [rbp-40]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-64]
mulsd xmm0, [rbp-72]
movsd [rbp-56], xmm0
mov rax, 0x4018000000000000
movq xmm0, rax
movsd [rbp-104], xmm0
movsd xmm0, [rbp-56]
divsd xmm0, [rbp-104]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-48]
leave
ret
MATLAB
function result = simpson_rule_single(x1, x2, a, b, c)
result = (((x2 - x1) * ((a + (4.0 * b)) + c)) / 6.0);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[x1_, x2_, a_, b_, c_] := (((x2 - x1) * ((a + (4.0 * b)) + c)) / 6.0);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Simpson's Rule Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/simpsons-rule-single-panel
MLA 9
MW SysArc. “Simpson's Rule Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/simpsons-rule-single-panel. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Simpson's Rule Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/calculus/simpsons-rule-single-panel.
Harvard
MW SysArc (2026) ‘Simpson's Rule Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/simpsons-rule-single-panel (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_simpson_rule_single_2026,
author = {{MW SysArc}},
title = {Simpson's Rule Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/simpsons-rule-single-panel},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Simpson's Rule Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/calculus/simpsons-rule-single-panel
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Simpson's rule do?
Approximate an integral from endpoint and midpoint function values.
How does the Simpson's rule work?
The calculator applies area≈(b−a)[f(a)+4f(mid)+f(b)]/6. A quadratic through three equally spaced samples integrates to Simpson's weighted formula.
What can I learn from the Simpson's rule?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .