8.3 Applications to Physics and Engineering In this section, we will discuss only one application of integral calculus to physics and engineering and this topic is: The Center of Mass of Planar Lamina Consider a thin flat plate of material with uniform density called a planar laminar. We think of center of mass as its balancing point.

Centroid of a Plane Region Consider a flat plate with uniform density that occupies a region R of the plane A. a R b x y Moment of R about the y-axis: Moment of R about the x-axis: Mass of Plate = (Density)(Area) Centroid of R: Note: If a lamina has the shape of a region that has an axis of symmetry, then the center of mass must lie on that axis.

B. R lies between on the interval [a, b] where x ab y Mass of R Centroid of R:

Examples: 1) Find the centroid of the region bounded by the curves. 1 e x y Solutions: D I

2) Find the center of mass of a semicircular plate of radius r. (-r, 0)(r, 0) By principle of symmetry, center of mass must lie on the y-axis. x y

3) Find the centroid bounded by the given curves. Points of intersection Solutions: (-2,0) (1, 3) x y