1. Disks, Washers and Shells Limerick Nuclear Generating Station, Pottstown, Pennsylvania Disk Method

2. Find the Volume of each solid

4. Cross Sections When a shaded region is the base of an object we use cross sections to find the volume Cross Sections are geometric shapes we have area formulas for that make integration and finding volume easy!

5. Cross Sections are like the slices of a loaf of bread The 3-Dimensional Solid has a flat side when dealing with cross sections!

6. Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve.

7. How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

8. The volume of each flat cylinder (disk) is: If we add the volumes, we get:

9. This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about a vertical axis, then the formula is: A shape rotated horizontally would be:

10. The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

11. The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.