MAT 21D Final: MAT 21D – Lecture 6 – Moments and Centers of Mass

J-1J, Jo Jo 2xy dx dy42. 2x dx dy In2 2 43. dx dy xy dx dy π/2 π /2 45. 6 sin (2x 3y) dx dy 46. dx 47-52. Evaluating integrals Evaluate the following integrals. A sketch is helpful. 47. 48. 49, sketch to n 』R 12yd: Rís bounded by y-2-x,y=Vx, and y=0. I Ry2 dA: Ris bounded by y=1.y=1-x, and y=x-1. 17x3xy dA: R is bounded by y=2-x, y=0, and x 4·-y2 in the first quadrant. d by 50. (x + y) dA: R is bounded by y = 1x1 and y = 4. 51. 1.3x2 dA: Ris bounded by y=0, y=2x+ 4, and y=x3. 52. 1 Rx2yd: R is bounded by y = 0, y = Vx, and y = x-2. 53-56. Volumes Use double integrals to calculate the volume of the following regions the 53. The tetrahedron bounded by the coordinate planes 54· The solid in the first octant bounded by the coordinate planes and 55。The segment of the cylinder x2 + y-1 bounded above by the 56. The solid beneath the cylinder y and above the region (x = 0, y = 0, z 0) and the plane z = 8-2x-4y the surface z 1-y-x plane z = 12 + x + y and below by: 0 x + 5

1. 10 points Use a double integral to find the volume of the tetahedron bounded coordinate planes and the plane 3x + 6y + 4x-12 = 0. 2. 10 points Evaluate dA where R is the triangular region with vertices (0,0), (1, 1), and (0, 1) 3. I0 points Use polar coordinates to evaluate the iterated integral (4 22)- dy dr 4. 20 points Find the center of mass of the lamina with mass density function 2 372 y 2 and which occupies the region bounded by the parabola y-2- and the line y = z.