The document discusses centroids and moments for planar laminae of uniform density. It defines a centroid as the center of mass and describes its properties, such as lying on lines of symmetry. The document presents equations to calculate the mass, moments about axes, and centroid of a lamina. It provides an example problem finding the centroid of a region bounded by two graphs. Finally, it discusses real-life applications of centroids for balancing objects, ensuring stability of structures, mapping crime densities, and increasing vehicle stability.

1. Centroids in a Planar LaminaYoung LeeBrian LauAlvin LauJonathan Javier

2. Introduction to CentroidsCentroid (Center of gravity)The center of mass of an object of uniform densityCharacteristics of a centroidIf a region has a line of symmetry, then its centroid is on that line of symmetryIf a region has two lines of symmetry, then the centroid is the point where they intersectMomentA measure of tendency to cause a body to rotate about a specific pointNo movement at the center of gravityPlanar LaminaA thin and flat plate of material of constant densityDensity: measure of mass per unit of area

3. Illustrationf(x)Rg(x)ab

4. IllustrationMass of Planar Lamina (R)

5. Illustrationf(x)𝜟xRf(x) — g(x)Subinterval, mig(x)ab

6. IllustrationMass of subinterval

7. Illustrationf(x)𝜟xRf(x) — g(x)g(x)ab

8. IllustrationMoment of mi about the x-axisMoment of mi about the y-axis

9. Centroidal EquationsMass of lamina:Moment of planar lamina about the x-axis:Moment of planar lamina about the y-axis: Center of Mass:

10. ExampleFind the centroid of the region bounded by the graphs f(x) = x and g(x) = x4Centroid: (0.56, 0.37)

11. Centroids: Real Life ApplicationBalanceBalancing thin objects on a point (2-D)Balancing two people on a see-saw (1-D)InfrastructureKnowing the centroid of parts of buildings ensures buildings to be stable (3-D)CrimesMapping the density of crimes in a certain areaLocomotionLocating the centroid of a vehicle to increase stability