Centroid X-Coordinate of a Region Calculator
The region under y = x from 0 to 2 has an area of 2 and balances at x = four thirds, right of the midpoint. Enter the curve's coefficients and the interval to get area, moment and centroid, worked exactly. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Coefficients
- 1, 0
- Lower bound
- 0
- Upper bound
- 2
- Area
- 2
- First moment y axis
- 2.6666666667
- X coordinate
- 1.3333333333
Enter the polynomial as coefficients in descending power order, together with the lower and upper bounds. The calculator evaluates the area integral, computes the first moment about the y-axis, and divides that moment by the area. It returns all three values so you can inspect the work, reuse intermediate results, and confirm the centroid rather than accepting an unexplained coordinate.
Represent the curve and interval correctly
Start by expressing the upper boundary as a polynomial f(x). Supply its coefficients from the highest power down to the constant term, keeping a zero wherever a power is missing. For example, x squared plus 3 is entered as [1, 0, 3], because the coefficient of x is zero.
Then provide a lower bound and an upper bound with the upper value strictly greater than the lower value. The intended geometric region lies between this curve and the x-axis across that interval, so the curve should remain above the axis. Before calculating, sketching the curve or checking important values can help confirm that your chosen interval describes one continuous region above the axis.
If the curve crosses the axis, split the problem at its roots and decide which bounded region you actually intend to measure instead of treating signed cancellation as ordinary geometric area.
Understand the area and moment calculation
The horizontal centroid coordinate is defined by x-bar equals M sub y divided by A. Here A is the area integral from the lower bound to the upper bound of f(x), and M sub y is the first moment about the y-axis, found by integrating x times f(x) over the same bounds. Multiplication by x gives material farther from the y-axis more influence, which is exactly what a balance coordinate must capture.
The solver integrates every polynomial term analytically: a term c times x to the n becomes c divided by n plus 1 times x to the n plus 1 for area, while the moment calculation first raises the term's power by one. Finally, it divides the accumulated moment by the accumulated area. Returning A and M sub y alongside x-bar makes it easy to compare the output with handwritten work and locate an incorrect coefficient or bound.
Interpret and verify the returned coordinate
Its position reflects how the region's area is distributed horizontally, not where the curve reaches its maximum and not the average of the two bounds. A symmetric region produces the midpoint, while extra area toward one side pulls the centroid in that direction. Units also provide a useful check.
If x is measured in meters and f(x) in meters, area is measured in square meters, the first moment is measured in cubic meters, and their quotient is measured in meters. The capability rejects reversed or equal bounds, non-finite inputs, numerical overflow, and a nonpositive signed area. A positive result alone does not prove the curve never crossed the axis, however.
When geometric validity matters, confirm that f(x) stays at or above zero throughout the selected interval before interpreting the ratio as the centroid of one region under the curve.
What you can do with it
Check a calculus exercise
Verify the area, first moment about the y-axis, and final horizontal centroid coordinate for a polynomial region.
Locate a lamina balance point
Find the horizontal balance coordinate of a uniform thin plate whose upper edge follows a polynomial curve.
Audit an engineering calculation
Expose both integrals and their quotient so a centroid used in later design work can be independently checked.
FAQ
What formula does the calculator use?
It uses x-bar = M_y / A, where A is the integral of f(x) and M_y is the integral of x f(x), both over the supplied interval.
How do I enter the polynomial?
Enter coefficients in descending power order and include zeros for missing powers. For example, x squared plus 3 is [1, 0, 3].
Must the curve stay above the x-axis?
Yes for the output to represent one ordinary region under the curve. The solver requires a positive integral but does not independently prove nonnegativity at every point.
Why are area and first moment returned?
They show the intermediate quantities behind the coordinate, making the result easier to verify and reuse.
Why is the centroid not at the middle of the interval?
Because it follows how the area is spread horizontally, not where the curve peaks or the average of the bounds. A symmetric region balances at the midpoint, while extra area on one side pulls the centroid toward it, as with y = x from 0 to 2.
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/centroid-region-x \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"coefficients":[1,0],"lower_bound":0,"upper_bound":2}'const res = await fetch("https://api.kit.forhosting.com/calculus/centroid-region-x", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"coefficients": [
1,
0
],
"lower_bound": 0,
"upper_bound": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/centroid-region-x",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"coefficients": [
1,
0
],
"lower_bound": 0,
"upper_bound": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/centroid-region-x", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"coefficients":[1,0],"lower_bound":0,"upper_bound":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"coefficients":[1,0],"lower_bound":0,"upper_bound":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/centroid-region-x", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"coefficients": [
1,
0
],
"lower_bound": 0,
"upper_bound": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.centroid_region_x",
"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 centroid x-coordinate calculator finds the horizontal balance point of a region bounded by a polynomial curve, the x-axis, and two vertical interval endpoints.
Represent the curve and interval correctly
This calculator does not parse formulas or discover intersections automatically; explicit coefficients and endpoints make the input deterministic and prevent ambiguous interpretations of notation.
Understand the area and moment calculation
It evaluates both antiderivatives at the endpoints and subtracts the lower value from the upper value.
Interpret and verify the returned coordinate
For a valid region with nonnegative height, the returned x-coordinate should fall between the interval endpoints. Use the returned area and first moment as verification data: dividing first_moment_y_axis by area should reproduce x_coordinate.
Technical questions
What does it cost?
The browser calculation is free. An API request costs $0.002.