Disks, Washers and Shells

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

2. 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.

3. The volume of each flat cylinder (disk) is: 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. In this case: r= the y value of the function thickness = a small change in x =dx

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

5. 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:

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

7. 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.

11. Washer Method

13. The volume of the washer is: The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. outer radius inner radius

14. The washer method formula is: This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.

15. r R If the same region is rotated about the line x=2: The outer radius is: The inner radius is:

16. Find the volume of the region bounded by , , and revolved about the y-axis. We can use the washer method if we split it into two parts: cylinder inner radius outer radius thickness of slice

17. Shell Method

20. Here is another way we could approach this problem: cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.

21. cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

22. This is called the shell method because we use cylindrical shells. cross section If we add all the cylinders from the smallest to the largest:

23. Find the volume of the solid obtained by rotating the region between y = x2 and y = x around the y axis.

24. Find the volume of the solid obtained by rotating the region between y = x2 + 4, y = 4, y = 8 and x = 0 around the y axis.

25. Find the volume of the solid obtained by rotating the region between y = x2 , y = 0, x = 1, and x = 2 around the line x = 4.

26. Disc v. Shell

27. When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method. p

28. Find the volume of the solid obtained by rotating the region between y = 2x2 - x3 and y = 0 around the y axis.

29. Find the volume of the solid obtained by rotating the region between y = 1/x, x = 1, x = 2 and y = 0 around the x axis.

30. Find the volume of the solid obtained by rotating the region between y = x3 + x + 1, y = 1 and x = 1 around the line x = 2.