1. What is the Centroid?

The centroid is the geometric center of a plane figure or the average position of all the points in the shape. For a 2D shape, it's the point where you could balance the shape perfectly on the tip of a pencil.

2. How Does the Calculator Work?

The calculator uses the centroid formula:

Where:

• \( A \) — Area of the shape (m²)
• \( y \) — Function describing the shape's boundary
• \( dx \) — Differential element along x-axis

Explanation: The formula integrates the function over the given limits and divides by the total area to find the average y-coordinate.

3. Importance of Centroid Calculation

Details: Centroid calculations are crucial in engineering and physics for determining balance points, center of mass, and structural analysis.

4. Using the Calculator

Tips: Enter the area of the shape, the function that describes its boundary, and the integration limits. The function should be in terms of x (e.g., "x^2" or "sin(x)").

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between centroid and center of mass?
A: Centroid is purely geometric, while center of mass considers density distribution. They coincide for uniform density.

Q2: How do I find centroid for complex shapes?
A: Break complex shapes into simpler components whose centroids you can calculate, then use composite centroid formulas.

Q3: Can this calculator handle parametric equations?
A: This version handles explicit functions y=f(x). Parametric equations would require a different approach.

Q4: What about shapes with holes?
A: Treat holes as negative areas in your calculations.

Q5: How accurate is this method?
A: The mathematical method is exact for the given function. Accuracy depends on correct input of the function and limits.