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Miss Battaglia AP Calculus

7- 2 Volume: The Disk Method (Day 2) Objective: Find volume of a solid revolution using the disk method and washer method. Miss Battaglia AP Calculus. The Disk Method.

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Miss Battaglia AP Calculus

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  1. 7-2 Volume: The Disk Method (Day 2)Objective: Find volume of a solid revolution using the disk method and washer method. Miss Battaglia AP Calculus

  2. 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 (formed by revolving a rectangle about an axis adjacent to one side of the rectangle. Volume of disk = (area of disk)(width of disk) = πR2Δx ΔV = πR2Δx

  3. The approximation of the volume of a solid becomes better as ||Δ||  0 (n  ∞). Volume of a solid =

  4. The Disk Method To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal Axis of RevolutionVertical Axis of Revolution

  5. The Washer Method The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.

  6. Example Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis. y = -x + 1

  7. Example Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

  8. Example Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

  9. Example Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis.

  10. Example Find the volume of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. (a) the x-axis (b) the y-axis (c) the line x = 3 (d) the line x = 6

  11. Classwork/Homework • Page 465 #15-21 odd (do evens for additional practice) 27, 33

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