Solids of Revolution Disk Method

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Presentation transcript:

Solids of Revolution Disk Method SECTION 7-2 Solids of Revolution Disk Method

The Disk Method If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk, which is formed by revolving a rectangle about an axis adjacent to one side of the rectangle Figure 7.13

The Disk Method To see how to use the volume of a disk to find the volume of a general solid of revolution, consider a solid of revolution formed by revolving the plane region in Figure 7.14 about the indicated axis. Figure 7.14

How do you find the Volume of a solid generated by revolving a given area about an axis? Slice the volume into many, many circular disks Then add up the volume of all the disks

Solids of Revolution: Disk Method The volume of a solid may be found by finding the sum of the disks. The volume of each circular disk is the area of a circle times the width of the disk. Volume is found by integration. The radius of each disk is the function for each value in the interval. The width is dx

Find Volume using Disk Method Revolve about a horizontal axis Slice perpendicular to axis – slices vertical Integrate in terms of x Revolve about a vertical axis Slice perpendicular to axis – slices horizontal Integrate in terms of y

Video clip on disk method http://www.youtube.com/watch?v=1CbZlM09zF8

Find Volume of a solid generated by revolving the given area about the x-axis 1) Consider the function on the interval [0,2]

1) (Continued) Find the volume of the solid 1) (Continued) Find the volume of the solid bounded by and the x-axis rotated about the x-axis on the interval [0,2]

2) Find the volume of the solid generated by revolving the region bounded by y = x – x2 and y = 0 about the x - axis 1 .25

3) Find the volume generated by revolving the region bounded by y = sec(x), and y = 0 about the x - axis

4) Find the volume generated by revolving the region bounded by about the y - axis Need in terms of x = ? Since revolution is about y

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of the following equation revolved about the y-axis: 5) Find the volume using the disk method with a horizontal disk. V=

Assignment Page 465 # 1-4, 7-10, 11a, 12b, 23, 25, 27, 31, and 33