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7.3 Volumes

7.3 Volumes. Disk and Washer Methods. 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. The volume of each flat cylinder (disk) is:.

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7.3 Volumes

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  1. 7.3 Volumes Disk and Washer Methods

  2. 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.

  3. The volume of each flat cylinder (disk) is: 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. In this case: r= the y value of the function thickness = a small change in x =dx

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

  5. 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:

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

  7. 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.

  8. The region bounded by a semicircle and its diameter shown below is revolved about the x-axis, which gives us a sphere of radius r. Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. Area of each cross section? (circle) What is y? dx –r r Equation of semicircle:

  9. Area of each cross section? (circle) Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. Volume: –r r (Remember r is just a number!!!)

  10. Volume: Example: Using calculus, derive the formula for finding the volume of a sphere of radius r. –r r

  11. Area of each cross section? (circle) Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2. What is r? r = 2 – x dy r Remember, we are using a dyhere!!! So, 2

  12. Area of each cross section? (circle) Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2. Volume: r 2 Bounds? From 0 to intersection (y-value!!!)

  13. The volume of the washer is: 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”. outer radius inner radius

  14. The washer method formula is: 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.

  15. r R If the same region is rotated about the line x=2: The outer radius is: The inner radius is:

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