Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 7.3 Volumes of rotation by Disks Limerick Nuclear Generating Station,

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Presentation transcript:

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, Volumes of rotation by Disks Limerick Nuclear Generating Station, Pottstown, Pennsylvania

Do you remember how the method to find the area of this region works? The area of each rectangle is: We can easily integrate this to get… heightbaseSum of all the rectangles Rectangles…oh yeah, I remember now.

Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. The units in this case are meters so the nose cone will have to be 4 meters long and 4 meters in diameter at its base. So I put a piece of wood in a lathe and turn it to a shape to match the curve. How much material will it take to make this nose cone? In other words, what will be the volume of this nose cone?

How could we find the volume of the cone? If the area of the region can be broken down into rectangles, then the volume of the cone can be cut into... The volume of each flat cylinder (disk) is: In this case: r = thickness = Flat cylinders…or disks the height of the rectangle dx

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

The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. We use a horizontal disk. 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.

on the interval  2 < x < 2. Find the area of the region bound by the curves y = 4 and y = x 2 A = h = 4 – x 2 upperlower A = Answer? dx height  base

on the interval  2 < x < 2 about the line y = 4. Find the volume of the solid generated by rotating the curve V = r = 4 – x 2 upperlower V = Answer?

 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: (height of the rectangle) 2 dx Where the height of the rectangle becomes the radius of the disk If the shape is rotated about the y-axis, then the formula is: (height of the rectangle) 2 dy