Centroid of a Curve Calculator

Analyze curves using numerical integration and derivatives. Review tables, moments, arc length, and centroid coordinates. Download reports, compare intervals, and study plotted curve behavior.

Calculator Input

Use x as the variable. Enter standard functions like sin(x), cos(x), sqrt(x+1), exp(x), log(x), and x^2+3x+1.

Use ordinary decimal numbers instead of scientific notation.

Curve Function y(x)

Lower Bound a

Upper Bound b

Integration Segments

Numerical Method

Decimal Places

Unit Label

Example Data Table

Example curve: y = x on the interval [0, 2]. This straight line has a centroid at the midpoint.

Function	Interval	Arc Length	x̄	ȳ	Notes
y = x	[0, 2]	2.828427	1.000000	1.000000	Line segment centroid equals the segment midpoint.
y = x	Derivative y′ = 1	ds/dx = √2	M_y = 2.828427	M_x = 2.828427	Symmetry keeps both centroid coordinates equal.

Formula Used

Arc length element: ds = √(1 + [y′(x)]²) dx

Total arc length: L = ∫_a^b √(1 + [y′(x)]²) dx

Moment about the x-axis: M_x = ∫_a^b y(x) √(1 + [y′(x)]²) dx

Moment about the y-axis: M_y = ∫_a^b x √(1 + [y′(x)]²) dx

Centroid coordinates: x̄ = M_y / L and ȳ = M_x / L

This page estimates y′(x) numerically and then evaluates the integrals with Simpson's rule or the trapezoidal rule.

How to Use This Calculator

Enter the curve as y(x). Use x as the variable.
Set the lower bound and upper bound for the interval.
Choose the number of segments. Larger values usually improve stability.
Select Simpson's rule for strong accuracy on smooth curves, or trapezoidal for a simpler estimate.
Pick the decimal precision and add a unit label if needed.
Submit the form. The result appears above the form and below the header.
Review the graph and the data table to validate the interval and function behavior.
Use the CSV or PDF buttons to keep a copy of the result.

FAQs

1. What does this calculator find?

It estimates the centroid of a plane curve treated as a thin wire. It also reports arc length and first moments about both axes.

2. Which variable should I use?

Use x as the independent variable. Write the function as y(x), such as x^2, sin(x), or sqrt(x+1).

3. Why does the calculator need derivatives?

The centroid of a curve depends on arc length weighting. Arc length uses ds = √(1 + [y′(x)]²) dx, so the derivative is required.

4. Which method should I choose?

Simpson's rule is usually more accurate for smooth curves. The trapezoidal rule is simpler and useful as a comparison check.

5. Why compare two numerical methods?

A small difference between methods suggests the estimate is stable. A larger difference means you may need more segments or a better interval choice.

6. Can I use negative intervals or negative y-values?

Yes. The calculator accepts negative bounds and negative function values. The centroid coordinates can also be negative, depending on the curve location.

7. What if my function fails on part of the interval?

The calculation stops when the function becomes non-finite or invalid. Adjust the interval so the function remains defined everywhere you evaluate it.

8. Is this for area centroids?

No. This page is for the centroid of the curve itself. Area centroids use different formulas based on area, not arc length.