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6.1 Area between two curves

6.1 Area between two curves.  A k = area of k th rectangle,  f( c k ) – g ( c k ) = height,  x k = width. Find the area of the region between the curves. Figure 4.23: When the formula for a bounding curve changes, the area integral changes to match. (Example 5).

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6.1 Area between two curves

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  1. 6.1 Area between two curves Ak = area of k th rectangle,f(ck) – g(ck ) = height, xk = width.

  2. Find the area of the region between the curves Figure 4.23: When the formula for a bounding curve changes, the area integral changes to match. (Example 5)

  3. Section 6.2 · Figure 5 Approximating the volume of a sphere with radius 1 (b) Using 10 disks, V » 4.2097 (c) Using 20 disks, V » 4.1940 (a) Using 5 disks, V » 4.2726 A

  4. 6. 2 Volumes – Solid of revolution Figure 5.6: The region (a) and solid (b) in Example 4. y = f(x) is rotated about x-axis on [a,b]. Find the volume of the solid generated. A cross-sectional slice is a circle and a slice is a disk.

  5. Volumes – Solid of revolution is rotated about the x-axis on [0, 4] Find the volume of the solid generated. Figure 5.6: The region (a) and solid (b) in Example 4.

  6. Volumes by disk-y axis rotation Find the volume of the solid generated by revolving a region between the y-axis and the curve x = 2/y from y = 1 to y = 4.

  7. Find the volume of the solid generated by revolving a region between the y-axis and the curve x = 2/y from y = 1 to 4.

  8. Washers If the region revolved does not border on or cross the axis of revolution, the solid has a hole in it.The cross sections perpendicular to the axis are washers. Figure 5.10: The cross sections of the solid of revolution generated here are washers, not disks, so the integral A(x) dx leads to a slightly different formula. b a V = Outside Volume – Inside Volume

  9. . The region bounded by the curve y = x2 +1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid of revolution.

  10. The inner and outer radii of the washer swept out by one slice. Outer radius R = - x + 3 and the inner radius r = x2 +1

  11. The inner and outer radii of the washer swept out by one slice. Outer radius R = - x + 3 and the inner radius r = x2 +1 Find the limits of integration by finding the x-coordinates of the points of intersection. x2 + 1= - x + 3 x2 + x –2=0 ( x+ 2 )(x – 1) = 0 x = -2 x = 1

  12. Calculation of volume Outer radius R = - x + 3 and the inner radius r = x2 +1

  13. y-axis rotation The region bounded by the parabola y = x2 and the line y = 2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid. Drawing indicates a dy integration so solve each equation for x as a function of y Set = to find y limits of integration y = 0 and y = 4 are limits

  14. The washer swept out by one slice perpendicular to the y-axis.

  15. calculation The region bounded by the parabola y = x2 and the line y = 2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid.

  16. 6. 3 Cylindrical Shells Figure 5.17: Cutting the solid into thin cylindrical slices, working from the inside out. Each slice occurs at some xk between 0 and 3 and has thickness x. (Example 1) Used to find volume of a solid of revolution by summing volumes of thin cylindrical shells or sleeves or tree rings.

  17. volume of a shell Imagine cutting and unrolling a cylindrical shell to get a (nearly) flat rectangular solid. Its volume is approximately V = length  height  thickness. ) Vshell =2(radius)(height)(thickness)

  18. problem The region enclosed by the x-axis and the parabola y = f(x) = 3x – x2 is revolved about the y – axis. Find the volume of the solid of revolution. Vshell =2(radius)(height)(thickness)

  19. The shell swept out by the kth rectangle. Notice this axis or revolution is parallel to the red rectangle drawn.

  20. problem The region bounded by the curve y = /x, , the x –axis and the line x = 4 is revolved about the y-axis to generate a solid. Find the volume of the solid.

  21. The region, shell dimensions, and interval of integration in

  22. The shell swept out by the rectangle in.

  23. Summary-Volumes-which method is best x-axis y-axis dy disk perpendicular dx dy shell parallel dx Axis of rotation

  24. Lengths of Plane curves Find the length of the arc formed by u = 1 + 4x du = 4dx du/4 = dx

  25. Follow the link to the slide. Then click on the figure to play the animation. A • Figure 6.2.5 • Figure 6.3.7 • Figure 6.2.12

  26. Section 6.3 · Figures 3, 4 Volumes by Cylindrical Shells

  27. Computer-generated picture of the solid in Example 9 Section 1 / Figure 1 A

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