1. What is a Centroid?

The centroid is the geometric center of a plane figure, the arithmetic mean position of all the points in the shape. For a 2D shape bounded by a function, it's calculated using calculus.

2. How Does the Calculator Work?

The calculator uses the centroid formula:

Where:

• \( \bar{y} \) — y-coordinate of centroid (length units)
• \( y \) — distance from axis
• \( dA \) — differential area element
• \( A \) — total area (square length units)

Explanation: The formula integrates the product of y and the differential area over the region, then divides by the total area.

3. Importance of Centroid Calculation

Details: Centroids are crucial in engineering and physics for determining balance points, centers of mass, and in structural analysis.

4. Using the Calculator

Tips: Enter the function defining the shape's boundary (in terms of x), and the lower and upper limits of integration. The function should be continuous over the interval.

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 you find the x-coordinate of centroid?
A: Similarly: \( \bar{x} = \frac{\int x \, dA}{A} \)

Q3: What if the shape has holes?
A: Treat them as negative areas in the integrals (composite bodies method).

Q4: Can this calculator handle parametric equations?
A: This version handles explicit functions y = f(x). Parametric forms require different integration.

Q5: What about polar coordinates?
A: Polar centroid calculations require converting to polar area elements (r dr dθ).