B) Determine the volume of the solid obtained by rotating the region bounded by : y= x -2x and y= x about the line y = 4 Solution First let's get the bounding region and the solid graphed. A W X Again, we are going to be looking for the volume of the walls of this object. Also since we are rotating about a horizontal axis we know that the cross-sectional area will be a function of x.Here are a couple of sketches of the boundaries of the walls of this object as well as a typical ring. The sketch on the left includes the back portion of the object to give a little context to the figure on the right. Now, we're going to have to be careful here in determining the inner and outer radius as they aren't going to be quite as simple they were in the previous two examples. Let's start with the inner radius as this one is a little clearer. First, the inner radius is NOT x. The distance from the x-axis to the inner edge of the ring is x, but we want the radius and that is the distance from the axis of rotation to the inner edge of the ring. So, we know that the distance from the axis of rotation to the x-axis is 4 and the distance from the x-axis to the inner ring is x. The inner radius must then be the difference between these two. 1) inner radius. 2) The outer radius is: 3) The cross-sectional area: A = It [ f2 - g2 ] outer inner A = n radius radius Using the formulas: Volume: = ]. A(x) dx V = [ A(y)dy 4) Find the volume of this solid