Question

Math232 2 Consider the region in first quadrant area bounded by y x,x=6, and the x-axis. Revolve this bounded region about the x-axis a) Sketch this region and find the volume of the solid of revolution; use the disk method, and show an element of the volume. (15 marks) b) Find the coordinates of the centroid of the solid of revolution Find the moment of inertia of the solid of revolution with respect to the x-axis. d)

Answer 1

Consider the curves y V, 6, y 0 about the x-axis y 0 The point of intersection of the two curves is as follows y 0 0 The volume of the curves about y0 given by disk method is as follows (())-(9() dr Volume T 6 =T r6 r dr = T 6 T 2 36 = T 2 - 18T

(b) dA M dy dr (V) da 6 - 0 - 4v6 For center-of-mass of the lamina We computed two integrals, one for each coordinate: r dA D dy dr 4V6 e6 Vdx 4v6 6 1 4V6 5 726 4V6 18 1 y dA JJ H fi y dy dr 46 dr 6 4V6 4V6 cle

3/3 4 2 18 33 Hence, the centre of mass is (,T)

5 (6, 2.449) (0, 0) (6, 0) (3.6, 0.919) 10 -5 -5 LO