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Section 7.2 – Volume: The Disk Method

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  1. Section 7.2 – Volume: The Disk Method

  2. White Board Challenge No Calculator Find the volume of the following cylinder: 6 ft 12 ft

  3. White Board Challenge No Calculator Calculate the volume V of the solid obtained by rotating the region between y = 5 and the x-axis about the x-axis for 1≤x≤7. 5 6

  4. Volumes of Solids of Revolution with Riemann Sums The Riemann Sum is set up by considering this cross sections of the solid (circles) each with thickness dx:

  5. Volumes of Solids of Revolution: Disk Method • Sketch the bounded region and the line of revolution. (Make sure an edge of the region is on the line of revolution.) • If the line of revolution is horizontal, the equations must be in y= form. If vertical, the equations must be in x= form. • Sketch a generic disk (a typical cross section). • Find the length of the radius and height of the generic disk. • Integrate with the following formula: Disk Method = No hole in the solid.

  6. Example 1 Calculate the volume of the solid obtained by rotating the region bounded by y = x2 and y=0about the x-axis for 0≤x≤2. Sketch a Graph Find the Boundaries/Intersections Radius = x2 Integrate the Volume of Each Generic Disk Line of Rotation Height = dx Make Generic Disk(s)

  7. Example 2 Calculate the volume of the solid obtained by rotating the region bounded by y = x2 and y=4about the line y = 4. Sketch a Graph Find the Boundaries/Intersections Radius = 4 - x2 Integrate the Volume of Each Generic Disk Line of Rotation NOTE: Because of the square, the order of subtraction does not matter. Height = dx Make Generic Disk(s)

  8. Example 3 Calculate the volume of the solid obtained by rotating the region bounded by y = x2,x=0, and y=4 about the y-axis. Sketch a Graph Since the Line of Revolution is Vertical, Solve for x Radius = √y We only need 0≤x≤2 x Find the Boundaries/Intersections Remember: 0≤x≤2 Integrate the Volume of Each Generic Disk Height = dy Make Generic Disk(s) Line of Rotation

  9. White Board Challenge No Calculator Find the volume of the following three-dimensional shape: 2ft 6 ft 12 ft

  10. White Board Challenge No Calculator Calculate the volume V of the solid obtained by rotating the region between y = 5 and the y =2 about the x-axis for 1≤x≤7. 5 2 6

  11. Area of a Washer The region between two concentric circles is called an annulus, or more informally, a washer: Rinner Router

  12. Volumes of Solids of Revolution: Washer Method Always a difference of squares. • Sketch the bounded region and the line of revolution. • If the line of revolution is horizontal, the equations must be in y= form. If vertical, the equations must be in x= form. • Sketch a generic washer (a typical cross section). • Find the length of the outer radius (furthest curve from the rotation line), the length of the inner radius (closest curve to the rotation line), and height of the generic washer. • Integrate with the following formula: Washer Method = Hole in the solid.

  13. Example 1 Calculate the volume V of the solid obtained by rotating the region bounded by y = x2and y=0 about the line y = -2 for 0≤x≤2. Sketch a Graph Find the Boundaries/Intersections Router = x2- -2 = x2+ 2 Rinner = 0- -2 = 2 Integrate the Volume of Each Generic Washer Line of Rotation Height = dx Make Generic Washer(s)

  14. Example 2 Calculate the volume V of the solid obtained by rotating the region bounded by y = exand y=√(x +2)about the line y = 2. Sketch a Graph Find the Boundaries/Intersections Router = 2 - ex Integrate the Volume of Each Generic Washer Line of Rotation Rinner = 2- √(x +2) Height = dx Make Generic Washer(s)

  15. “Warm-up”: 1985 Section I No Calculator CAN DO NOW:

  16. Volume of a Right Solid A right solid is a geometric solid whose sides are perpendicular to the base. The volume of a right solid is the area of the base times the height. HSolid ABase

  17. Volumes of Solids: Slicing Method • Sketch the bounded region. • If the cross section is perpendicular to the x-axis, the equations must be in y= form. If the y-axis, the equations must be in x= form. • Sketch a generic slice (a typical cross section). • Find the area of the base and the height of the generic slice. • Integrate with the following formula: Must Answer #1: What does the length across the bounded region represent in your generic slice? Must Answer #2: How does the length across the bounded region help find the area of the base of the generic slice?

  18. Example 1 Find the volume of the solid created on a region who base is bounded by y = √x and the x-axis for 0≤x≤9. Let each cross section be perpendicular to the x-axis and be a square. Sketch a Graph Find the Boundaries/Intersections Height = dx ABase = ASquare =side2 Integrate the Volume of Each Generic Slice = (√x)2 Side Length Make Generic Slice(s)

  19. Example 2 Find the volume of the solid created on a region who base is bounded by x2 + y2 = 1. Let each cross section be perpendicular to the x-axis and be a squares with diagonals in the xy-plane. Sketch a Graph Since the Cross Sections are Per. to the x-axis, solve for y ABase = ASquare =side2 Height = dx If diameter is known, a side length is… Find the Boundaries/Intersections Diagonal Integrate the Volume of Each Generic Slice Make Generic Slice(s)

  20. White Board Challenge Calculator A solid has base given by the triangle with vertices (-4,0), (0,8), and (4,0). Cross sections perpendicular to the y-axis are semi-circles with diameter in the plane. What is the volume of the solid? Height = dy Diameter Radius = -½y+4 ABase = ½πr2