Volume: Disk and Washer Methods 6.2 Volume: Disk and Washer Methods LIMERICK GENERATING STATION Limerick Generating Station, located in Limerick Township, Montgomery County, PA, is a two-unit nuclear generation facility capable of producing enough electricity for over 1 million homes. The plant site is punctuated by two natural-draft hyperbolic cooling towers, each 507 feet tall, which help cool the plant. Limerick's two boiling water reactors, designed by General Electric, are each capable of producing 1,143 net megawatts. Unit 1 began commercial operation in February 1986, with Unit 2 going on-line in January 1990. Limerick Nuclear Generating Station, Pottstown, Pennsylvania Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

Objectives Find the volume of a solid of revolution using the disk method. Find the volume of a solid of revolution using the washer method. Find the volume of a solid with known cross sections.

3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. h s This correlates with the formula: dh 3

Method of Slicing: 1 Sketch the solid and a typical cross section. Find a formula for V(x). (Note that I used V(x) instead of A(x).) 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.

Calculus in Motion Demos

h 45o x A 45o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. x y If we let h equal the height of the slice then the volume of the slice is: x h 45o Since the wedge is cut at a 45o angle: Since

Even though we started with a cylinder, p does not enter the calculation! x y

Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections

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.

r= the y value of the function 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: In this case: r= the y value of the function thickness = a small change in x = dx

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

This application of the method of slicing is called the disk method 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 the x-axis, then the formula is: A shape rotated about the y-axis would be:

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

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.

and is revolved about the y-axis. Find the volume. 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”. The volume of the washer is: outer radius inner radius

This application of the method of slicing is called the washer method 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: Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.

Homework Handout 8.1 #1-27 odd (skip #7) #41, 46, 48 p