AP Calculus BC. Find a formula for the area. B. The cross sections are squares with bases in the xy -plane.
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1 AP Calculus BC Find a formula for the area Homework Problems Section 7. Ax 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 to the semicircle y 1 x. 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 times the length of its sides.) D. The cross sections are equilateral triangles with bases in the xy -plane.
2 Calculus BC Problems 7..docx. Find the volume of the solid that lies between planes perpendicular to the x -axis at x and x 4. The cross sections perpendicular to the axis on the interval 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 to the semicircle y 1 x. 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.
3 Calculus BC Problems 7..docx 1. Find the volume of the solid generated by revolving the region bounded by y 9 x and y about the x -axis. 14. Find the volume of the solid generated by revolving the region bounded by y x x and y about the x -axis. 15. Find the volume of the solid generated by revolving the region bounded by y x, y 1, and x about the x -axis. 16. Find the volume of the solid generated by revolving the region bounded by y x, y x, and x 1 about the x -axis. 17. Find the volume of the solid generated by revolving the region bounded by y x 1 and y x about the x -axis.
4 4 Calculus BC Problems 7..docx 18. Find the volume of the solid generated by revolving the region bounded by y 4 x and y x about the x -axis. 19. Find the volume of the solid generated by revolving the region bounded by y sec x and y on the interval about the x -axis. x 4 4. Find the volume of the solid generated by revolving the region bounded by y x, y, and x about the x -axis. 9. Find the volume of the solid generated by revolving the region bounded by y x and the lines y and x about A. the x -axis. B. the y -axis. C. the line y. D. the line x 4
5 Calculus BC Problems 7..docx 5. Find the volume of the solid generated by revolving the triangular region bounded by y x, y and x 1 about A. the line x 1 B. the line x. 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 8 y. 5
6 6 E. the line y 5 Calculus BC Problems 7..docx 7. Use the shell method to find the volume of the solid generated by revolving the region bounded by y x, y, and x 4 about the y -axis. 9. Find the volume of the solid if the base of the solid is the region between the curve y sinx x -axis. The cross sections perpendicular to the x -axis are A. equilateral triangles with bases running from the x -axis to the curve. and the interval, on the B. squares with bases running from the x -axis to the curve.
7 Calculus BC Problems 7..docx 7 4. Find the volume of the solid if the solid lies between planes perpendicular to the x -axis at x and x. The cross sections perpendicular to the x -axis are A. circular disks with diameters running from the curve y tanx to the curve y sec x. B. squares whose bases rund form the curve y tanx 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 and y. The cross sections perpendicular to the y -axis are circular disks with diameters running from the y -axis to the parabola x 5y. 4. Find the volume of the solid if the base of the solid is the disk x y 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 8 Calculus BC Problems 7..docx sin x, x 48. Let f x x. 1, x B. 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. 6. TRUE OR FALSE The volume of a solid of a known integrable cross section area Justify your answer. Ax from x a to x b b is A x dx. a 64. TRUE OR FALSE If the region enclosed by the y -axis, the line y, and the curve y x is revolved about the y -axis, the volume of the solid is given by the definite integral y dy. Justify your answer. 65. The base of a solid S is the region enclosed by the graph of y lnx, 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..718 B C..718 D..171 E. 7.88
9 Calculus BC Problems 7..docx 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. 6. B C. 5.4 D E Let R be the region enclosed by the graph of y x, 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. 18 C. 56 D. 6 E Let R be the region enclosed by the graphs of y e x x, y e, and the x 1. Which of the following gives the volume of the solid generated when R is revolved about the x -axis? 1 x x A. e e x x dx B. e e x x dx C. 1 x x x x D. e e dx E. 1 1 e e dx 1 e e dx
10 1 Calculus BC Problems 7..docx QUICK QUIZ FOR AP PREPERATION: SECTIONS You may use a graphing calculator to solve the following problems 1 1. The base of a solid is the region in the first quadrant bounded by the x -axis, the graph of y sin 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..117 B..85 C..467 D..571 E Let R be the region in the first quadrant bounded by the graph of y x x 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. A. x 1 x x dx B. x 1 x x dx C. D. x x dx E. 1 x x dx x x x dx r t e. A developing country consumes oil at a rate given by. t million barrels per year, where t is time measured in years, for t 1. Which of the following gives the amount of oil consumed by the country during the time interval t 1? 1 A. r 1 B. r 1 r C. r ' t dt D. r t dt E. 1 r Let R be the region bounded by the graphs of y x, A. Find the area of R. y e x, and the y -axis. 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.
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