5th Period: AP Calculus

1. Centroids in a Planar Lamina Young Lee Brian Lau Alvin Lau Jonathan Javier

2. Introduction to Centroids Centroid (Center of gravity) The center of mass of an object of uniform density Characteristics of a centroid If a region has a line of symmetry, then its centroid is on that line of symmetry If a region has two lines of symmetry, then the centroid is the point where they intersect Moment A measure of tendency to cause a body to rotate about a specific point No movement at the center of gravity Planar Lamina A thin and flat plate of material of constant density Density: measure of mass per unit of area

3. Illustration f(x) R g(x) a b

4. Illustration Mass of Planar Lamina (R)

5. Illustration f(x) 𝜟x R f(x) — g(x) Subinterval, mi g(x) a b

6. Illustration Mass of subinterval

7. Illustration f(x) 𝜟x R f(x) — g(x) g(x) a b

8. Illustration Moment of mi about the x-axis Moment of mi about the y-axis

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

10. Example Find the centroid of the region bounded by the graphs f(x) = x and g(x) = x4 Centroid: (0.56, 0.37)

11. Centroids: Real Life Application Balance Balancing thin objects on a point (2-D) Balancing two people on a see-saw (1-D) Infrastructure Knowing the centroid of parts of buildings ensures buildings to be stable (3-D) Crimes Mapping the density of crimes in a certain area Locomotion Locating the centroid of a vehicle to increase stability