1. Calculus 6-R Unit 6 Applications of Integration Review Problems

2. 1 Find the area of the region bounded by the graphs of y = x 2 - 4x and y = x - 4. Find the area of the region bounded by the graphs of y = -x 2 + 2x + 3 and y = 3.

3. Review Problems 2 Find the area of the region bounded by the graphs of f(x) = 6x - x 2 and g(x) = x 2 - 2x Find the area of the region bounded by the graphs of f(x) = x3 - 2x and g(x) = -x

4. Review Problems 3 Find the area of the region bounded by the graphs of f(x) = x 3 - 6x and g(x) = -2x 8 Find the area of the region bounded by the graphs of f(x) = x 3 + x 2 - 12x and g(x) = -x 2 + 3x

5. Review Problems 4 Find the area of the region bounded by the graphs of f(x) = x 3 + x 2 - 6x and g(x) = -x 2 + 2x Find the area of the region bounded by the graphs of f(x) = x 3 + x 2 - 72x and g(x) = -x 2 + 8x

6. Review Problems 5 Find the area of the region bounded by y = 1/x and 2x + 2y = 5 Find the area of the region bounded by the graphs of f(x) = sin x and g(x) = cos x, for π/4 ≤ x ≤ 5π/4

7. Review Problems 6 Find the area of the region bounded by the graphs of y = x 3 - 6x 2 + 8x and y = 0 8 Find the area of the region bounded by the graph of y 2 = x 2 - x 4

8. Review Problems 7 Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 3, y = 1, and x = 2 about the x-axis Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 3, x = 2, and y = 1 about the y-axis

9. Review Problems 8 Find the volume of the solid formed by revolving the region bounded by the graphs about the x-axis π Find the volume of the solid formed by revolving the region bounded by the graphs of y = -x 2 + 1 and y = 0 about the x-axis.

10. Review Problems 9 Find the volume of the solid formed by revolving the region bounded by the graphs about the x-axis Find the volume of the solid formed by revolving the region bounded by the graphs of y = 4x 2 and y = 16 about the line y = 16

11. Review Problems 10 Find the volume of the solid formed by revolving the region bounded by the graph of y 2 = x 4 (1 - x 2 ) about the x-axis Find the volume of the solid formed by revolving the region bounded by the graphs of y = 0, and x = 6 about the x-axis. 8π8π

12. Review Problems 11 Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2, and x = 2 about the x-axis Find the volume of the solid formed by revolving the region bounded by the graphs of y = 2x 2, x = 0, and y = 2 about the y-axis. π

13. Review Problems 12 Find the volume of the solid formed by revolving the region bounded by the graphs of y - 4x - x 2 and y = 0 about the x-axis. π Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 + 3, y = 3, x = 0 and x = 2 about the line y = 3 π

14. Review Problems 13 Find the volume of the solid formed by revolving the region bounded by the graphs of y = 3 - x 2 and y = 2 about the line y = 2 π Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 + 3, y = 3, x = 0 and x = 2 about the line y = 3 π

15. Review Problems 14 Find the volume of the solid formed by revolving the region bounded by y = x 3, y = 1, and x = 2 about the x-axis π Find the volume of the solid formed by revolving the region bounded by y = e x, y = 0, x = 0, and x = 1 about the y-axis 2π2π

16. Review Problems 15 Find the volume of the solid formed by revolving the region bounded by y = 1, and x = 4 about the x-axis π Find the volume of the solid formed by revolving the region bounded by y = x 2 and y = 4 about the x-axis π

17. Review Problems 16 Find the volume of the solid formed when the graph of the region bounded by and y = 1 is revolved about the x-axis Find the volume of the solid formed when the graph of the region bounded by y = e x, y = 1, and x = 4 is revolved about the x-axis

18. Review Problems 17 Find the volume of the solid formed when the graph of the region bounded by f(x) = 3x 2 and g(x) = 5x + 2 about the x-axis 180.924 Find the volume of the solid formed when the graph of the region bounded by f(x) = -3x 2 + 8 and g(x) = 3x 2 + 2 about the x-axis 251.327

19. Review Problems 18 The base of a solid is the region bounded by the graphs of x = y 2 and x = 4. Each cross section perpendicular to the x- axis is a triangle of altitude 2. Find the volume of the solid. Find the volume of the solid formed when the graph of the region bounded graphs of y = x 2, y = 1, and x = 0. about the line y=2

20. Review Problems 19 Find the volume of the solid generated by revolving the region bounded by the graphs of x = 6, and the x-axis about the y-axis. Find the volume of the solid generated by revolving the region bounded by the graphs of x = 2, and the x-axis about the y-axis

21. Review Problems 20 Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 and y = 4 about the x-axis. Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = -x + 2, y = 0, and x = 0 about the y-axis.

22. Review Problems 21 Find the volume of the solid generated when the region bounded by the graphs of y = x 3, y = 0, and x = 1 is revolved about the line x = 2. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 2x 2 + 4x and y = 0 about the y-axis

23. Review Problems 22 Find the volume of the solid formed by revolving the region bounded by the graphs of and y = 2 about the y-axis Consider the region bounded by the graphs of y = x 2, and x = 2. Calculate the volume of the solid formed when this region is revolved about the line x = 2. 2π2π

24. Review Problems 23 The base of a solid is the region bounded by the graphs of y = 2 sin x and x = 4. Each cross section perpendicular to the x-axis is a semicircle. Find the volume. A round hole is drilled through the center of a spherical solid of radius r. The resulting cylindrical hole has a height of 4 cm. What is the volume of the solid that remains.

25. Review Problems 24 The base of a solid is the region bounded by the graphs of x 2 + y 2 ≤ 1 The cross sections by planes perpendicular to the y-axis between y = -1 and y = 1 are isosceles right triangles with one leg in the disk. The solid lies between planes perpendicular to the y-axis and y = 0 and y = 2. The cross sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola

26. Review Problems 25 The base of a solid is the region between the curve and the interval [0,π] on the x-axis. The cross sections perpendicular to the x-axis are equilateral triangles with bases running from the x- axis to the curve. The base of a solid is the region between the curve and the interval [0,π] on the x-axis. The cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve. 8

27. Review Problems 26 The base of a solid is the region between the x-axis and the inverted parabola y = 4 – x 2. The vertical cross sections of the solid perpendicular to the y-axis are semicircles. Compute the volume of the solid The base is the semicircle, where -3 ≤ x ≤ 3. The cross sections perpendicular to the x-axis are squares. Find the volume 36

28. Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 8 8 π 8π8π π ππ ππ π 2π2π

29. Answers 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. π π 180.924251.327 2π2π 8 36