Centroid of the Semicircular Region We start by reviewing moments and centers of mass from single-variable calculus. Consider a flat p late ( called a l amina) i n t he x y-plane w ith u niform d ensity ρ t hat o ccupies a region R of the plate. Suppose that R is bounded by the curve y = f (x). The moment of a region R about the y-axis is: Rb Pn My = limn→∞ i=1 ρ x̄i f (x̄i ) △ x = ρ a xf (x) dx. The moment of a region R about the x-axis is: Rb Pn Mx = limn→∞ i=1 ρ 12 [f (x̄i )]2 △ = ρ a 1 2 [f (x)]2 dx. The center of mass of the plate (or the centroid of R) is located at the point (x̄, ȳ), where x̄ = Rb xf (x) dx Rab f (x) dx a ȳ = Rb 1 a R b2 a [f (x)]2 dx . f (x)dx Exercise 1. Find the centroid of the semicircular region bounded by the x-axis on the graph of √ y = a2 − x2 with a > 0. Class Exercise 1. Find the centroid of the region bounded by the given curves. √ (a) y = x, y = 0, and x = 4 (b) y = sin x, y = 0, 0 ≤ x ≤ π (c) y = 2 − x2 and y = x (d) y = x3 , x + y = 2, and y = 0 Formula: Suppose a lamina occupies a region D of the xy-plane and its density at point (x, y) in D is given by ρ(x, y), where ρ is a continuous function on D. Then, the total mass m of the lamina is RR Pk Pl ∗ m = limk,l→∞ i=1 j=1 ρ(x∗ij , yij )△A= ρ(x, y) dA. D Definition: The moment of the entire lamina about the x-axis: RR Pm Pn ∗ ∗ Mx = limm,n→∞ i=1 j=1 yij y · ρ(x, y) dA. ρ(x∗ij , yij )△A= D Definition: The moment of the entire lamina about the y-axis: RR Pm Pn ∗ My = limm,n→∞ i=1 j=1 x∗ij ρ(x∗ij , yij )△A= x · ρ(x, y) dA. D Formula: The coordinates (x̄, ȳ) of the center of mass of a lamina occupying the region D and having density function ρ(x, y) are RR RR M 1 1 x x̄ = my = m x · ρ(x, y) dA, ȳ = M y · ρ(x, y) dA, m = m D D where the mass m is given by m= RR ρ(x, y) dA. D Exercise 2. A lamina has the shape of an isosceles right triangle with equal sides of length a. The area mass density at a point P is directly proportional to the square of the distance from P to the vertex that is opposite the hypotenuse. Find the center of mass. (Swok Sec 17.6 Ex 1) Exercise 3. A lamina has the shape of the region R in the xy-plane bounded by the parabola x = y 2 and the line x = 4. The area mass density at the point P (x, y) is directly proportional to the distance from the y-axis to P . Find the center of mass. (Skow Sec 17.6 Ex 2) Class Exercise 2. Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. (#4,6,8,10) (a) D = { (x, y) | 0 ≤ x ≤ a, 0 ≤ y ≤ b }; ρ(x, y) = 1 + x2 + y 2 (b) D is the triangular region enclosed by the lines x = 0, y = x, and 2x + y = 6; ρ(x, y) = x2 (c) D is bounded by y = x2 and y = x + 2 ; ρ(x, y) = kx √ (d) D is bounded by the parabolas y = x2 and x = y 2 ; ρ(x, y) = x Definition: The moment of inertia of a particle of mass m about an axis is defined to be mr2 , where r is the distance from the particle to the axis. Definition: The moment of inertia of the lamina about the x-axis is RR 2 Pm Pn ∗ 2 ∗ Ix = limm,n→∞ i=1 j=1 (yij ) ρ(x∗ij , yij )△A= y ρ(x, y) dA. D Definition: The moment of inertia of the lamina about the y-axis is RR 2 Pm Pn ∗ Iy = limm,n→∞ i=1 j=1 (x∗ij )2 ρ(x∗ij , yij )△A= x ρ(x, y) dA. D Definition: The moment of inertia of the lamina about the origin is RR 2 Pm Pn ∗ 2 ∗ ) ] ρ(x∗ij , yij )△A= (x + y 2 ) ρ(x, y) dA. Io = limm,n→∞ i=1 j=1 [(x∗ij )2 + (yij D Exercise 4. A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first q uadrant. The plate’s density at the p oint (x, y ) is ρ(x, y ) = 6x + 6 y + 6. Find the plate’s moments of inertia about the coordinate axes and the origin. (Hass Sec 15.6 Ex 4) Class Exercise 3. Suppose a lamina occupies the part of the disk x2 + y 2 ≤ 1 in the first quadrant and the density at any point is proportional to the square of its distance from the origin. Find the moments of inertia Ix , Iy , I0 for the lamina. (#18) Class Exercise 4. Suppose a lamina occupies with a triangle (0, 0), (b, 0), and (0, h) with constant density ρ(x, y) = ρ. Find the moments of inertia Ix and Iy . (#22) Class Exercise 5. Suppose a lamina occupies the region under the curve y = sin x from x = 0 to x = π with constant density ρ(x, y) = ρ. Find the moments of inertia Ix and Iy . (#24)