Moment of Inertia Integral Calculator

Calculator Inputs

Moment type

Strip function

Use explicit multiplication, such as 2*x, sqrt(x), or sin(x).

Scale factor c

Set 1 for a pure geometric second moment.

Lower bound

Upper bound

Integration steps

Higher even values usually improve Simpson accuracy.

Plot sample points

Displayed decimals

Unit label

Example: cm, m, or unit.

Formula Used

For a strip model, the second moment is built from a distance-squared weighting. When the region is described by a vertical strip of height h(x), the calculator evaluates Iᵧ = c∫ x²h(x) dx. When the region is described by a horizontal strip of width b(y), it evaluates Iₓ = c∫ y²b(y) dy.

The auxiliary geometric relations are A = ∫ strip d(variable) and centroid = (∫ variable·strip d(variable)) / A. The radius of gyration is reported as k = √(I / A) for the unscaled geometric second moment.

The numerical engine uses Simpson’s rule with an automatic even step count. A trapezoid estimate and a refined Simpson pass are also computed, which helps you compare stability and judge numerical accuracy.

How to Use This Calculator

Select whether you want the second moment about the x-axis or y-axis.
Enter the strip function. Use h(x) for y-axis work or b(y) for x-axis work.
Set lower and upper bounds for the interval being integrated.
Leave the scale factor at 1 for a standard area moment, or change it for weighted problems.
Choose the number of integration steps and display precision.
Press Calculate to show results above the form, review the graph, and export CSV or PDF files.

Example Data Table

Example region	Axis	Strip function	Bounds	Expected second moment	Comment
Parabolic strip	y-axis	h(x) = x²	0 to 2	Iᵧ = ∫₀² x⁴ dx = 6.4	Good first validation case for a curved height function.
Linear taper	x-axis	b(y) = 4 - y	0 to 3	Iₓ = ∫₀³ y²(4-y) dy = 6.75	Represents a simple triangular-style width variation.
Semicircle height model	y-axis	h(x) = sqrt(9 - x²)	0 to 3	Numeric result from integration	Useful when an exact antiderivative is inconvenient.

FAQs

1. What does this calculator measure?

It evaluates a second moment integral, often called an area moment of inertia. The result shows how strongly a planar region is distributed away from a chosen reference axis.

2. What is the difference between Iₓ and Iᵧ?

Iₓ weights the strip by its distance from the x-axis using y². Iᵧ weights the strip by its distance from the y-axis using x².

3. Why must I type multiplication explicitly?

The parser reads algebra safely and clearly. Writing 2*x instead of 2x avoids ambiguity and improves evaluation reliability for numerical integration.

4. What does the scale factor do?

It multiplies the computed integral by a constant. Leave it at 1 for a geometric result, or change it when your model needs a density or weighting factor.

5. Why are Simpson and trapezoid results both shown?

They give a quick consistency check. When both values are close and the estimated error is small, the numeric result is usually stable for the chosen interval and step count.

6. Can I use trigonometric and root functions?

Yes. Supported functions include sin, cos, tan, sqrt, log, exp, and related standard functions, as long as they stay finite on the interval.

7. What if the strip function becomes negative?

The calculator still integrates it as a signed function. That is mathematically valid, but geometric quantities such as area and centroid no longer represent an ordinary physical region.

8. When should I increase the step count?

Increase steps for rapidly changing curves, oscillating functions, or long intervals. More steps usually improve accuracy, though extremely difficult functions may still need careful interpretation.