Midpoint Riemann sum calculator
With four equal strips, the midpoint sum for x^2 from 0 to 1 is 0.328125, each strip 0.25 wide. Enter the interval, the number of strips and the polynomial's coefficients to approximate any definite integral. Free in your browser.
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Worked example
- A
- 0
- B
- 1
- N
- 4
- Coefficients
- 1, 0, 0
- Polynomial
- x^2
- Delta x
- 0.25
- Approximation
- 0.328125
A midpoint Riemann sum is the standard first numerical estimate of a definite integral when you partition an interval into equal-width rectangles and evaluate the height of each rectangle at the midpoint of its base. Students meeting the definition of the definite integral, tutors grading approximation tables, and scripts that need a reproducible integral check without a full quadrature library all need the same contract: choose a and b, choose how many subintervals n, state the integrand, and receive M_n with Δx spelled out.
What a midpoint Riemann sum computes
A Riemann sum approximates that area by slicing [a, b] into n contiguous pieces and replacing the graph on each piece with a horizontal strip whose height is a sample of f. With equal width, the common step is Δx equals (b minus a) divided by n. The i-th subinterval runs from a plus i times Δx to a plus (i plus one) times Δx for i from zero through n minus one, and its midpoint is a plus (i plus one half) times Δx.
The midpoint sum is then the sum over i of f at that midpoint times Δx. When f is a polynomial given by coefficients highest degree first, each sample is evaluated with Horner arithmetic so the result is deterministic floating-point arithmetic with no symbolic engine. If b is less than a, Δx is negative and the sum inherits the orientation of the integral, matching the signed-area convention used in calculus courses.
When a equals b the width is zero and the approximation is zero for any finite polynomial values.
Where midpoint sums come in handy
Calculus classrooms and online homework systems use a midpoint Riemann sum calculator so learners can check table rows for arbitrary polynomials without arithmetic slips on the midpoints. Exam generators need a deterministic oracle that rejects n equals zero instead of silently returning nonsense, and that matches the same rounding teachers show in worked solutions. Failed validation does not consume balance.
Keep n modest for interactive demos, pin the approximation field in tests, and you get reproducible midpoint sums for LMS plugins, tutoring bots, and edge workers without standing up symbolic or CAS infrastructure of your own.
What you can do with it
Classroom midpoint tables
Let students enter a, b, n, and polynomial coefficients and verify M_n and Δx against textbook midpoint Riemann sum tables.
Exam and LMS auto-grading
Score approximation items with a deterministic midpoint oracle that rejects invalid n and never invents sample points.
Notebook integral checks
Sanity-check ∫_a^b of a polynomial load or density model with equal subintervals before calling heavier quadrature.
FAQ
What happens if n is zero or not an integer?
The midpoint rule needs at least one equal-width subinterval, so n must be a positive integer up to 100000.
Can a be greater than b?
Yes. Δx becomes negative and the approximation inherits the orientation of the definite integral, matching the signed-area convention.
How do I calculate a midpoint Riemann sum?
Enter the two ends of the interval, the number of equal strips and the polynomial's coefficients from the highest power down. For x^2 from 0 to 1 with 4 strips, each is 0.25 wide and the sum is 0.328125.
Why use midpoints instead of the left or right edges?
Sampling at the center of each strip tends to cancel much of the error that left or right edge samples leave on smooth functions, so with the same number of rectangles the midpoint estimate is usually closer.
How do I enter the function?
Type the polynomial's coefficients from the highest power to the constant term. For x^2, enter 1, 0, 0; for 2x + 1, enter 2, 1. Only polynomials in one variable are accepted.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/calculus/riemann-sum-midpoint \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":0,"b":1,"n":4,"coefficients":[1,0,0]}'const res = await fetch("https://api.kit.forhosting.com/calculus/riemann-sum-midpoint", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 0,
"b": 1,
"n": 4,
"coefficients": [
1,
0,
0
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/riemann-sum-midpoint",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 0,
"b": 1,
"n": 4,
"coefficients": [
1,
0,
0
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/riemann-sum-midpoint", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":0,"b":1,"n":4,"coefficients":[1,0,0]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":0,"b":1,"n":4,"coefficients":[1,0,0]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/riemann-sum-midpoint", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 0,
"b": 1,
"n": 4,
"coefficients": [
1,
0,
0
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.riemann_sum_midpoint",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This midpoint Riemann sum calculator accepts the endpoints of [a, b], a positive integer n, and a univariate polynomial written as highest-degree-first coefficients, then forms M_n = Δx times the sum of f at each midpoint. Invalid n, empty coefficients, or non-finite endpoints fail with invalid_input and are not billed.
What a midpoint Riemann sum computes
The definite integral of a continuous function f from a to b is the signed area under the graph of y equals f of x between those endpoints. The midpoint rule picks the sample at the center of each subinterval, which for smooth functions tends to cancel leading-order trapezoid error better than pure left or right endpoints on the same mesh.
Input contract, response shape, and validation
POST a JSON body with four required pieces: a and b as finite numbers for the integration interval; n as a positive integer between one and one hundred thousand for the number of equal-width subintervals; and coefficients as a non-empty array of finite numbers highest degree first for the polynomial integrand, with aliases coeffs and poly accepted. Missing fields, empty coefficient lists, non-finite values, and non-integer or non-positive n fail with invalid_input naming the problem. On success the response echoes a, b, n, and the cleaned coefficients; a compact polynomial string such as x^2 or 2x+1 for display; delta_x as the common subinterval width; approximation as the midpoint sum rounded for stable cross-runtime goldens; method set to the literal midpoint; and a formula string restating M_n equals Δx times the sum of f at a plus (i plus one half) Δx. Example: a equals zero, b equals one, n equals four, coefficients one zero zero for f of x equals x squared yields delta_x equals zero point two five and approximation zero point three two eight one two five, which sits next to the exact integral one third. Pair the numeric result with your own plot of the rectangles if you need a visual; this endpoint returns the sum and the mesh width, not SVG or step-by-step prose beyond the formula field.
Where a midpoint Riemann sum API earns its keep
Engineering notebooks often need a quick integral check for a polynomial load or density model before wiring a heavier adaptive quadrature routine; calling a tiny parse API avoids importing a full numerical stack for a four-line midpoint loop.
Technical questions
How do I compute a midpoint Riemann sum with this tool?
Send finite endpoints a and b, a positive integer n, and highest-degree-first coefficients for the polynomial f. The API returns Δx = (b−a)/n and M_n = Δx · Σ f(a+(i+1/2)Δx) for i = 0..n−1.
What form does the integrand take?
A univariate polynomial as a non-empty coefficients array highest degree first (aliases coeffs, poly). Example [1,0,0] means f(x)=x²; [2,1] means f(x)=2x+1.
What does the midpoint Riemann sum calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.