Discs and washers: Volume of solids _ 2 Determine the volume of the solid obtained by rotating the region bounded by y x _ 435+ 5 X = 1.x = 4 and the xandyaxes: Step 1: The rst thing to do is get a sketch of the bounding region and the solid obtained by rotating the region about the x- axis. Here are both of these sketches. estate I a .x x a I IDuuhu Okay, to get a cross section we cut the solid at any x. Below are a couple of sketches showing a typical cross section. The sketch on the right shows a cut away of the object with a typical cross section without the caps. The sketch on the le shows just the curve we're rotating as well as its mirror image along the bottom of the solid. 9 8 1 6 5 l 3 2 l V-KLMS '/3\\ ' x s r \\ In this case the radius is simply the distance from the x-axis to the curve and this is nothing more than the function value at that particular x as shown above. The cross-sectional area is then, A(x)= 7r(x2 4x+ 5)2 =7; (x' 8f +26x' 40x+25) a) Find the volume of the solid above: V = f A(x) dx