4: Due Thursday, October 15 at 8:55 am in class. 1) Consider the region R bounded by the curves y = 0, x = 0, x = 3, and 9x y = x2 +9 . a) Find the area of R. b) Find the volume of the solid obtained by rotating R about the y axis. c) Find the volume of the solid obtained by rotating R about x = 1. d) Find the volume of the solid obtained by rotating R about y = 1. 2) Find the volume of the following solid S: the base of S is the unit disk x2 y 2 1. The cross-sections of S by planes perpedicular to the x axis, between y = 1 and y = 1, are equilateral triangles with one side in the disk. 3) Consider the region R above the x axis, to the right of x = 1, and below 1 y = x . Fully justify your answers. a) Does R have nite area? If yes, nd it. b) Does the solid obtained by rotating R about the x axis have nite volume? If yes, nd it. 4) In this problem, f (x) is a nonnegative continuous function with f (x) = 0 for x 1. The volume of the solid obtained by revolving the region bounded by y = f (x), the x axis, x = 0, and x = a about the x axis is denoted by V (a), for each a > 0. a) Check that V (1) = 0 and and compute V (1). b) Say that V (a) = (a 1)3 for a > 1. Find f