The center of mass is the balance point of an object or region, where its mass can be treated as if it were concentrated at one point. For a flat region with uniform density, this point is called the centroid. Calculus lets us find centroids of curved shapes by slicing the region into many tiny pieces and adding their contributions.
This matters in physics, engineering, design, and geometry because balance and rotational behavior depend on where mass is distributed.
For a plane region, each small area element contributes both area and moment. The centroid coordinates are found by dividing the total moment about an axis by the total area. When density is constant, the density cancels, so the centroid depends only on shape.
Symmetry can greatly simplify the work because the centroid must lie on any line of symmetry.
Key Facts
- For uniform density, centroid and center of mass are the same point.
- Area under y = f(x) from x = a to x = b is A = integral from a to b of f(x) dx.
- For a region under y = f(x), x-bar = (1/A) integral from a to b of x f(x) dx.
- For a region under y = f(x), y-bar = (1/A) integral from a to b of (1/2)[f(x)]^2 dx.
- For a region between y = f(x) and y = g(x), A = integral from a to b of [f(x) - g(x)] dx.
- For a region between curves, y-bar = (1/A) integral from a to b of (1/2)([f(x)]^2 - [g(x)]^2) dx.
Vocabulary
- Center of mass
- The point where an object's mass can be considered concentrated for analyzing translation and balance.
- Centroid
- The geometric center of a plane region or solid when density is uniform.
- Moment
- A measure of how strongly area or mass is distributed relative to an axis.
- Area element
- A small piece of area, often written as dA, used to build an integral over a region.
- Symmetry axis
- A line that divides a shape into matching halves and must contain the centroid when the shape is uniform.
Common Mistakes to Avoid
- Forgetting to divide by total area: Moment integrals give total weighted area, not the centroid coordinate by themselves.
- Using the wrong moment formula: x-bar uses the moment about the y-axis, while y-bar uses the moment about the x-axis.
- Ignoring the lower curve in a region between curves: The height is f(x) - g(x), not just the top function f(x).
- Assuming the centroid is always inside the shaded region by visual guessing alone: The correct location comes from moments, and visual estimates can be misleading for irregular shapes.
Practice Questions
- 1 Find the centroid of the region under y = 4 from x = 0 to x = 6. Compute A, x-bar, and y-bar.
- 2 Find the centroid of the region under y = x^2 from x = 0 to x = 2. Use A = integral from 0 to 2 of x^2 dx, x-bar = (1/A) integral from 0 to 2 of x^3 dx, and y-bar = (1/A) integral from 0 to 2 of (1/2)x^4 dx.
- 3 A uniform region is symmetric about the vertical line x = 3 but has an uneven top boundary. What can you conclude about x-bar, and why does symmetry not automatically determine y-bar?