1. Volumes using washers

2. Now that you have successfully designed a 4 by 4 meter nose cone, your boss brings to you a larger nose cone that is 16 meters long by 8 meters wide. In order to safely ship it, you need to cover it with a layer of insulation exactly as thick as shown in the gray area below: What you know about the design is that the curvature of the nose cone and the insulation can be given by…

3. Now that you have successfully designed a 4 by 4 meter nose cone, your boss brings to you a larger nose cone that is 16 meters long by 8 meters wide. In order to safely ship it, you need to cover it with a layer of insulation exactly as thick as shown in the gray area below: How many cubic meters of insulation will it take to make this layer?

4. The disks now have holes in them, making them more like washers. If we use vertical slices Rotate the region about the x-axis We find the volume of a washer by considering it to be two disks; an outer and inner disk. We find the volume by subtracting the volume of the hole from the volume of the disk.

5. When we subtract, we get… The volume of the disk is R 2 R r The volume of the hole is r 2 outer radius inner radius dx

6. The region bounded by and is revolved about the y-axis. Find the volume. The volume of the washer is: outer radius inner radius

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

8. 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. The washer method formula is: Where R is the distance from the axis of rotation to the outer curve and r is the distance from the axis of rotation to the inner curve.

9. The washer method formula is: Where R is the distance from the axis of rotation to the outer curve and r is the distance from the axis of rotation to the inner curve. Since the region being rotated is one bound by two curves, could we also consider R to be the upper curve and r to be the lower curve? R r

10. 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: outer radius inner radius thickness of slice Cylinder First, the cylinder on the bottom Everything above that will require washers…

11. outer radius inner radius thickness of slice Cylinder