Mathematics · Calculus
Trapezoidal Rule Calculator
Approximate an integral from two endpoint function values.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Width=2−0=2.
- Average height=(1+5)÷2=3.
- Estimate=2×3=6.
Understand Trapezoidal rule
One idea, three depths
Choose how deeply to explain Trapezoidal rule
Trapezoidal rule: Approximate an integral from two endpoint function values.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Trapezoidal rule to answer this question: approximate an integral from two endpoint function values? Enter Start a, End b, f(a), and 1 other input; the calculator shows Trapezoidal estimate. For example: From x=0 to 2 with endpoint heights 1 and 5, area≈6. The answer tells you Trapezoidal estimate.
Age 15Explain it to a 15-year-oldConnect it to the formula
A straight line between endpoints forms a trapezoid approximating the curve. The rule is area≈(b−a)[f(a)+f(b)]/2. Its input values are Start a, End b, f(a), f(b), and the main result is Trapezoidal estimate. For example: From x=0 to 2 with endpoint heights 1 and 5, area≈6.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated trapezoidal rule relation over the valid real-number domain stated below. The implemented relation is area≈(b−a)[f(a)+f(b)]/2, evaluated from Start a, End b, f(a), f(b) to produce Trapezoidal estimate. A straight line between endpoints forms a trapezoid approximating the curve. One interval may be crude for strongly curved functions.
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(b) must be a finite real number.
Important boundary: One interval may be crude for strongly curved functions.
The formula
area≈(b−a)[f(a)+f(b)]/2
How the calculator works through it
It substitutes Start a, End b, f(a), f(b) into the formula and exposes every numerical step above. The main output is Trapezoidal estimate, accompanied by Interval width, Average height.
Read the result correctly
The Trapezoidal estimate is the direct answer to “approximate an integral from two endpoint 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
From x=0 to 2 with endpoint heights 1 and 5, area≈6.
Where this model stops being reliable
One interval may be crude for strongly curved functions.
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 Trapezoidal rule works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Trapezoidal rule uses area≈(b−a)[f(a)+f(b)]/2. 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 Trapezoidal rule.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Trapezoidal 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(b).
- Evaluate the principal relationship: area≈(b−a)[f(a)+f(b)]/2.
- Return Trapezoidal estimate and check the domain conditions described above.
Python
from math import *
def trapezoidal_rule_single(x1, x2, a, b) -> float:
return (((x2 - x1) * (a + b)) / 2.0)
assert abs(trapezoidal_rule_single(0, 2, 1, 5) - 6) < 1e-6 * max(1.0, abs(6))
C
#include <assert.h>
#include <math.h>
double trapezoidal_rule_single(double x1, double x2, double a, double b) {
return (((x2 - x1) * (a + b)) / 2.0);
}
int main(void) {
const double expected = 6;
const double actual = trapezoidal_rule_single(0, 2, 1, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double trapezoidal_rule_single(double x1, double x2, double a, double b) {
return (((x2 - x1) * (a + b)) / 2.0);
}
int main() {
constexpr double expected = 6;
const double actual = trapezoidal_rule_single(0, 2, 1, 5);
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 trapezoidal_rule_single(double x1, double x2, double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global trapezoidal_rule_single
section .text
trapezoidal_rule_single:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd xmm0, [rbp-16]
subsd xmm0, [rbp-8]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-24]
addsd xmm0, [rbp-32]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-48]
divsd xmm0, [rbp-72]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
leave
ret
MATLAB
function result = trapezoidal_rule_single(x1, x2, a, b)
result = (((x2 - x1) * (a + b)) / 2.0);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[x1_, x2_, a_, b_] := (((x2 - x1) * (a + b)) / 2.0);
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). Trapezoidal Rule Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval
MLA 9
MW SysArc. “Trapezoidal Rule Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Trapezoidal Rule Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval.
Harvard
MW SysArc (2026) ‘Trapezoidal Rule Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_trapezoidal_rule_single_2026,
author = {{MW SysArc}},
title = {Trapezoidal Rule Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Trapezoidal Rule Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/calculus/trapezoidal-rule-single-interval
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Trapezoidal rule do?
Approximate an integral from two endpoint function values.
How does the Trapezoidal rule work?
The calculator applies area≈(b−a)[f(a)+f(b)]/2. A straight line between endpoints forms a trapezoid approximating the curve.
What can I learn from the Trapezoidal 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 .