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AP Calculus BC Homework Problems Section 7.3 Find a formula for the area A x of the cross sections of the solid that are perpendicular to the x -axis. 1. The solid lies between the planes perpendicular to the x -axis at x 1 and x 1 . The cross sections perpendicular to the x -axis between these planes run from the semicircle y 1 x 2 to the semicircle y 1 x 2 . A. The cross sections are circular disks with diameters in the xy -plane. B. The cross sections are squares with bases in the xy -plane. C. The cross sections are squares with diagonals in the xy -plane. (The length of a square’s diagonal is its sides.) D. The cross sections are equilateral triangles with bases in the xy -plane. 2 times the length of 2 Calculus BC Problems 7.3.docx 3. Find the volume of the solid that lies between planes perpendicular to the x -axis at x 0 and x 4 . The cross sections perpendicular to the axis on the interval 0 x 4 are squares whose diagonals run from y x to y x . 5. Find the volume of the solid that lies between planes perpendicular to the x -axis at x 1 and x 1 . The cross sections perpendicular to the x -axis between these planes are squares whos bases run from the semicircle y 1 x 2 to the semicircle y 1 x 2 . 7. Find the volume of the solid generated by revolving the shaded region about the x -axis. 9. Find the volume of the solid generated by revolving the shaded region about the y -axis. Calculus BC Problems 7.3.docx 3 13. Find the volume of the solid generated by revolving the region bounded by y 9 x and y 0 about the x -axis. 2 14. Find the volume of the solid generated by revolving the region bounded by y x x 2 and y 0 about the x -axis. 15. Find the volume of the solid generated by revolving the region bounded by y x , y 1 , and x 0 about the x -axis. 16. Find the volume of the solid generated by revolving the region bounded by y 2x , y x , and x 1 about the x -axis. 17. Find the volume of the solid generated by revolving the region bounded by y x2 1 and y x 3 about the x -axis. 4 Calculus BC Problems 7.3.docx 18. Find the volume of the solid generated by revolving the region bounded by y 4 x2 and y 2 x about the x -axis. 19. Find the volume of the solid generated by revolving the region bounded by y sec x and y 2 on the interval 4 x about the x -axis. 20. Find the volume of the solid generated by revolving the region bounded by y x , y 2 , and x 0 about the x -axis. 29. Find the volume of the solid generated by revolving the region bounded by y x and the lines y 2 and x 0 about… A. the x -axis. C. the line y 2 . B. the y -axis. D. the line x 4 4 Calculus BC Problems 7.3.docx 30. Find the volume of the solid generated by revolving the triangular region bounded by y 2x , y 0 and x 1 about… A. the line x 1 B. the line x 2 33. Use the shell method to find the volume of the solid generated by revolving the shaded region about… A. the x -axis. B. the line y 1 . C. the line y 8 . 5 5 6 Calculus BC Problems 7.3.docx E. the line y 2 5 37. Use the shell method to find the volume of the solid generated by revolving the region bounded by y x , y 0 , and x 4 about the y -axis. 39. Find the volume of the solid if the base of the solid is the region between the curve y 2 sin x and the interval 0, on the x -axis. The cross sections perpendicular to the x -axis are… A. equilateral triangles with bases running from the x -axis to the curve. B. squares with bases running from the x -axis to the curve. Calculus BC Problems 7.3.docx 40. Find the volume of the solid if the solid lies between planes perpendicular to the x -axis at x 7 3 and x 3 . The cross sections perpendicular to the x -axis are… A. circular disks with diameters running from the curve y tan x to the curve y sec x . B. squares whose bases rund form the curve y tan x to the curve y sec x . 41. Find the volume of the solid if the solid lies between planes perpendicular to the y -axis at 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 x 5y 2 . 42. Find the volume of the solid if the base of the solid is the disk x2 y2 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. 8 Calculus BC Problems 7.3.docx sin x , 48. Let f x x 1, B. 0 x . x 0 Find the volume of the solid generated by revolving the shaded region about the y -axis. STANDARDIZED TEST QUESTIONS You may use a graphing calculator to solve the following problems. b 63. TRUE OR FALSE The volume of a solid of a known integrable cross section area A x from x a to x b is A x dx . a Justify your answer. 64. TRUE OR FALSE If the region enclosed by the y -axis, the line y 2 , and the curve y x is revolved about the y -axis, the 2 volume of the solid is given by the definite integral y 2 dy . Justify your answer. 0 65. The base of a solid S is the region enclosed by the graph of y ln x , the line x e , and the x -axis. If the cross sections of S perpendicular to the x -axis are squares, which of the following gives the best approximation of the volume of S ? A. 0.718 B. 1.718 C. 2.718 D. 3.171 E. 7.388 Calculus BC Problems 7.3.docx 9 3 2 66. Let R be the region in the first quadrant bounded by the graph of y 8 x , the x -axis, and the y -axis. Which of the following gives the best approximation of the volume of the solid generated when R is revolved about the x -axis? A. B. 60.3 C. 115.2 D. 225.4 E. 319.7 361.9 67. Let R be the region enclosed by the graph of y x 2 , the line x 4 , and the x -axis. Which of the following gives the best approximation of the volume of the solid generated when R is revolved around the y -axis? A. 64 B. 128 C. 256 D. E. 360 512 68. Let R be the region enclosed by the graphs of y e x , y e x , and the x 1 . Which of the following gives the volume of the solid generated when R is revolved about the x -axis? 1 A. e x e x dx 1 B. 0 e2 x e 2 x dx 0 2x e 2 x dx 0 1 D. e 1 E. ex e 0 1 C. e 0 x 2 dx x e x dx 2 10 Calculus BC Problems 7.3.docx QUICK QUIZ FOR AP PREPERATION: SECTIONS 7.1-7.3 You may use a graphing calculator to solve the following problems 1. The base of a solid is the region in the first quadrant bounded by the x -axis, the graph of y sin1 x , and the vertical line x 1 . For this solid, each cross section perpendicular to the x -axis is a square. What is the volume? A. B. 0.117 C. 0.285 0.467 D. E. 0.571 1.571 2. Let R be the region in the first quadrant bounded by the graph of y 3x x 2 and the x -axis. A solid is generated when R is revolved about the vertical line x 1 . Set up, but do not evaluate, the definite integral that gives the volume of this solid. 3 A. 3 2 2 x 1 3x x dx B. D. 2 3x x 2 2 C. 1 0 3 3 2 2 x 1 3x x dx 2 x 3x x dx 2 0 3 dx E. 0 3x x dx 2 0 3. A developing country consumes oil at a rate given by r t 20e0.2t million barrels per year, where t is time measured in years, for 0 t 10 . Which of the following gives the amount of oil consumed by the country during the time interval 0 t 10 ? A. r 10 B. r 10 r 0 10 C. r 't dt 0 10 D. r t dt E. 10 r 10 0 4. Let R be the region bounded by the graphs of y x , y e x , and the y -axis. A. Find the area of R . B. Find the volume of the solid generated when R is revolved about the horizontal line y 1 . C. The region R is the base of a solid. For this solid, each cross section perpendicular to the x -axis is a semicircle whose diameter runs from the graph of y x to the graph of y e x . Find the volume of this solid.