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The disk method

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Information about The disk method
Education

Published on February 17, 2014

Author: roneick

Source: slideshare.net

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AP Calculus Warm up A cylinder has a height of 9 feet and a volume of 706.5 cubic feet. Find the radius of the cylinder. Use 3.14 for π .

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, as shown in Figure 7.13. Figure 7.13 2

The Disk Method The volume of such a disk is Volume of disk = (area of disk)(width of disk) = πR2w where R is the radius of the disk and w is the width. 3

7.3 day 2 Disk and Washer Methods Limerick Nuclear Generating Station, Pottstown, Pennsylvania Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington

y= x 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. →

Lathe

y= x 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: π r 2 ⋅ the thickness π ( x) 2 dx In this case: r= the y value of the function thickness = a small change in x = dx →

y= x The volume of each flat cylinder (disk) is: π r 2 ⋅ the thickness π ( x) 2 dx If we add the volumes, we get: ∫ π( 4 x 0 ) 2 dx 4 = ∫ π x dx 0 4 π 2 = x 2 0 = 8π →

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.

The Disk Method This approximation appears to become better and better as So, you can define the volume of the solid as Volume of solid = Schematically, the disk method looks like this. 10

The Disk Method A similar formula can be derived if the axis of revolution is vertical. Figure 7.15 11

Example 1 – Using the Disk Method Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 ≤ x ≤ π) about the x-axis. Solution: From the representative rectangle in the upper graph in Figure 7.16, you can see that the radius of this solid is R(x) = f(x) 12 Figure 7.16

Example 1 – Solution cont’d So, the volume of the solid of revolution is 13

Example 2 – Revolving about a line that is not the coordinate axis.  Find the volume of the solid formed by revolving the f ( x) = 2 − x 2 and g ( x) = 1 about the region bounded by: line: y = 1 14

Example 1: (Use Graphing Calculator) Find the volume of the solid formed by revolving the region bounded by the graph of f ( x) = .5 x 2 + 4 and the x-axis, between x = 0 and x = 3, about the x-axis. Example 2: (No calculator) Rotate the region below About the y- axis. Example 3: (Use technology) rotate the region Bounded by the x2 Graphs of y = 2 , and f ( x) = 4 − about the line y = 2 4 15

1 The region between the curve x = y , 1 ≤ y ≤ 4 and the y-axis is revolved about the y-axis. Find the volume. y 1 We use a horizontal disk. x 1 3 1 = .707 2 1 = .577 3 4 1 2 2 The thickness is dy. dy The radius is the x value of the 1 function = . y 2  1  V =∫ π dy  y  1   4 =∫ π 4 1 1 dy y volume of disk 0 = π ln y 1 = π ( ln 4 − ln1) 4 = π ln 22 = 2π ln 2 →

y 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: 500 ft x x = .000574 y 2 − .439 y + 185 The volume can be calculated using the disk method with a horizontal disk. π∫ 500 0 ( .000574 y 2 − .439 y + 185 ) dy ≈ 24, 700, 000 ft 3 2 →

y = 2x y = x2 The region bounded by y = x 2 and y = 2 x is revolved about the y-axis. Find the volume. If we use a horizontal slice: y = 2x y =x 2 y = x2 y=x The “disk” now has a hole in it, making it a “washer”. The volume of the washer is:  V =∫ π 0   4 ( y) 2  y −  2 2   dy   1 2  V = ∫ π  y − y  dy 0 4   4 V =π∫ 4 0 1 2 y − y dy 4 ( π R − π r ) ⋅ thickness π ( R − r ) dy 2 2 2 2 outer radius 4 1  1 = π  y2 − y3  12  0 2 inner radius  16  = π 8 −  3  8π = 3 →

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: b V = π ∫ R 2 − r 2 dx a Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula. →

y = x2 y = 2x r y = 2x y =x 2 y = x2 y=x r = 2− y y2 = π ∫ 4 − 2 y + − 4 + 4 y − y dy 0 4 1 4 1 2 = π ∫ −3 y + y + 4 y 2 dy 0 4 4 V = π ∫ R 2 − r 2 dy 0 2 ( y  =π ∫ 2−  − 2− y 0 2  ) 2 dy ( ) 4  3 2 1 3 8  = π ⋅ − y + y + y  12 3 0  2 16 64  8π  = π ⋅  −24 + +  = 3 3 3  3 2  y2  = π ∫  4 − 2 y +  − 4 − 4 y + y dy 0 4   4 The outer radius is: y R = 2− 2 The inner radius is: R 4 4 If the same region is rotated about the line x=2: π

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