The centroid of a region is the geometric center or the average position of all the points in a shape. It is often denoted by coordinates $(\bar{x}, \bar{y})$ in a 2D plane.
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The centroid can be found using the formulas $\bar{x} = \frac{1}{A} \int_a^b x f(x) \, dx$ and $\bar{y} = \frac{1}{A} \int_a^b \frac{1}{2} [f(x)]^2 \, dx$, where $A$ is the area of the region.
Centroids are used to determine the center of mass for planar regions with uniform density.
For composite shapes, the centroid can be found by dividing the shape into simpler parts, finding their centroids, and then taking a weighted average based on their areas.
In symmetrical objects, the centroid lies along the axis of symmetry.
If a region is bounded by two curves, you may need to use double integrals to find its centroid.
Review Questions
What are the formulas for finding the coordinates of a centroid in a given region?
How would you determine the centroid of a composite shape?
Why is understanding centroids important in calculating centers of mass?
Related terms
Moment: The moment is a measure of the tendency of a force to rotate an object about an axis. Mathematically, it involves integrating distance times density over an area or volume.
Center of Mass: The point at which all mass of an object can be considered to be concentrated for purposes of analyzing translational motion.
Area: The measure of space within a two-dimensional boundary, often used as part of calculating centroids and moments.