Solids-of-revolution-Disk-and-Ring-Method.pptx

solids of revolution disk and ring method

1. Solids of revolution Disk Method

2. The volume of a solid bounded by planes is easily defined in solid geometry. However, for a solid bounded by curved surfaces or by a combination of curved surfaces and planes we may obtain the volume by evaluating a definite integral. Provided, the area of every plane section parallel to some fixed plane can be expressed as a function of its distance from the latter. A solid of revolution is the figure formed when a plane region is revolved about a fixed line. The fixed line is called the axis of revolution . For short, we shall refer to the fixed line as axis or AOR . The volume of a solid of revolution may be calculated using the following methods: DISK , RING and SHELL METHOD Volume of a Solid

3. This method is used when the element (representative strip) is perpendicular to and touching the axis. Meaning, the axis is part of the boundary of the plane area. When the strip is revolved about the axis of rotation a DISK is generated. A. DISK METHOD:   To find the volume of the entire solid :  

4. A. DISK METHOD:      

5. Sketch the bounded region and the line of revolution. ( Make sure an edge of the region is on the line of revolution .) If the line of revolution is horizontal, the equations must be in form. If vertical, the equations must be in form. Sketch a generic disk (a typical cross section). Find the length of the radius and height of the generic disk. Integrate with the following formula: Disk Method = No hole in the solid   A. DISK METHOD:  

6. Calculate the volume of the solid obtained by rotating the region bounded by y = x 2 and y= about the x -axis for 0 ≤ x ≤ 2 . Example 1 Axis of Rotation y =   x = 2 x = 0 y = 0

7. Calculate the volume of the solid obtained by rotating the region bounded by y = x 2 , x= 0, and y= 4 about the y -axis. Example 2 Axis of Rotation x = 2 x = 0 y = 4 y = 0

8. 11 Find the volume of the solid generated when the region enclosed by , and is revolved about the -axis.   Example 3

9. Ring or Washer method is used when the element (or representative strip) is perpendicular to but not touching the axis . Since the axis is not a part of the boundary of the plane area, the strip when revolved about the axis generates a ring or washer. B. RING or WASHER METHOD:    

10. B. RING or WASHER METHOD:    

11. Area of a Washer: The region between two concentric circles is called an annulus, or more informally, a washer: B. RING or WASHER METHOD:     R inner R outer  

12. Sketch the bounded region and the line of revolution. If the line of revolution is horizontal, the equations must be in y = form. If vertical, the equations must be in x = form. Sketch a generic washer (a typical cross section). Find the length of the outer radius (furthest curve from the rotation line), the length of the inner radius (closest curve to the rotation line), and height of the generic washer. Integrate with the following formula: B. RING or WASHER METHOD:     Washer Method = Hole in the solid

13. Calculate the volume V of the solid obtained by rotating the region bounded by and about the line for .   EXAMPLE 4

14. EXAMPLE 5

15. Determine the volume of the solid generated by the region between and , revolved about the -axis.   EXAMPLE 6 y = 4x y =