Using the fact that the centroid of a triangle lies at the intersection of the triangle's medians, which is the point that lies one-third of the way from each side toward the opposite vertex, find the centroid of the triangle whose vertices are (0,0), (5,0), and (0,7). The centroid of the triangle is (x,y) , where x = and y =. (Type integers or simplified fractions.)Using the fact that the centroid of a triangle lies at the intersection of the triangle's medians, which is the point that lies one-third of the way from each side toward the opposite vertex, find the centroid of the triangle whose vertices are (0,0), (a,0), and (0,b). Assume a > 0 and b > 0. The centroid of the triangle is (x.y) , where x = and y =].Find the centroid of the thin plate bounded by the graphs of g(x) = x" and f(x) = - x + 30. Use the equations shown below with 8 = 1 and M = area of the region covered by the plate. b *= - 8x[f(x) - g(x)] dx The centroid of the thin plate is (x.y) , where x = and y = (Type integers or simplified fractions.)Find the centroid of the thin plate bounded by the graphs of the functions g(x) = x (x - 1) + 2 and f(x) = x + 2 with 8= 3 and M = mass of the region covered by the plate. Use the following equations. 8xIf(X) - 9(x)]dx a a The centroid of the thin plate is (x,y) , where x = and y = (Type integers or simplified fractions.)The square region with vertices (8,0), (16,8), (8,16), and (0,8) is revolved about the x-axis to generate a solid. Find the volume and surface area of the solid. The volume of the solid generated is cubic units. (Type an exact answer, using it as needed.)Find the volume of the torus generated by revolving the circle [x 4)2 + 3,3 = 9 about the y-axis. The volume of the torus generated is D cubic units. {Type an exact answer: using 11: as needed.)