Find the volume of a solid with regular cross sections whose base is the region between two functions

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1 Area Volume Big Ideas Find the intersection point(s) of the graphs of two functions Find the area between the graph of a function and the x-axis Find the area between the graphs of two functions Find the volume of a solid with regular cross sections whose base is the region between two functions Find the volume when a region is revolved around a line Disk/Washer method Shell method is not required but allowed Find the equation of a line that divides an area into two equal parts Calculator or non-calculator in recent years (9 11) has been on non-calculator part (1 was on calculator part again) Awarding of points on the exam is weighted toward set-up. Mike Koehler 8-1 Area Volume

2 Mike Koehler 8 - Area Volume

3 Mike Koehler 8 - Area Volume

4 Mike Koehler 8-4 Area Volume

5 Multiple Choice Questions Volumes Cross Section ab 86 calc 1 The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = tan x, the horizontal line y =, and the vertical line x = 1. For this solid, each cross section perpendicular to the x- axis is a square. What is the volume of the solid? A).561 B) 6.61 C) 8.46 D) E).77 bc 89 calc The region bounded by the graph of y = x x and the x-axis is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid? A) 1. B) 1.67 C).577 D).46 E).67 98bc 86 calc The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x+ y = 8, as shown in the figure on the right. If cross sections of the solid perpendicular to the x- axis are semicircles, what is the volume of the solid? A) B) C) D) 67.1 E) ab 84 calc The base of a solid S is the region in the first quadrant enclosed by the graphs of y = ln x, the line x = e, and the x-axis. If the cross section of S perpendicular to the x-axis are squares, then the volume of S is A) 1 B) 1 e C) 1 D) E) ( 1) Mike Koehler 8-5 Area Volume

6 97bc 87 The base of a solid is the region in the first quadrant enclosed by the graph of y = x and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume of the solid is given by A) ( ) B) ( ) y dy y dy C) ( ) x dx D) ( ) x dx E) ( ) x dx 88bc 5 The base of a solid is the region in the first quadrant enclosed by the parabola y = 4x, the line x = 1, and the x-axis. Each plane section of the solid perpendicular to the x-axis is a square. The volume of the solid is A) B) C) D) E) bc 9 The base of a solid is the region enclosed by of y = e x, the coordinate axes, and the line x =. If all plan cross sections perpendicular to the x-axis are squares, then its volume is ( 1 e 6 ) A) 1 6 B) e 6 C) e D) e E) 1 e Mike Koehler 8-6 Area Volume

7 Volumes Revolution 97ab 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 generated is A) B) C) D) E) bc 77 When the region enclosed by the graphs of the solid generated is given by A) ( x x ) dx B) ( 4 ) C) ( x x ) dx D) ( x x ) dx E) ( x x ) dx ( ) x x x dx y x y x x = and = 4 is revolved about the y-axis, the volume of 9ab Let R be the region in the first quadrant enclosed by the graph of y = ( x+ 1 ) 1, the line x = 7, the x-axis, and the y-axis. The volume of the solid generated when R is revolved about the y-axis is given by A) 7 ( ) 7 B) x( 1 x) 1 dx + C) ( + ) D) ( ) 1 7 y 1 dy x x dx E) ( ) 1+ 1 x dx 9ab The region enclosed by the x-axis, the line x =, and the curve y = x is rotated about the x -axis. What is the volume of the solid generated? 9 A) B) C) D) 9 E) 6 5 Mike Koehler 8-7 Area Volume

8 9bc19 calc. The shaded region R, between the curve y = kx x and the x-axis, shown in the figure to the right, is rotated about the y-axis to form a solid whose volume is 1 cubic units. Of the following, which best approximates k? A) 1.51 B).9 C).49 D) 4.18 E) bc What is the volume of the solid generated by rotating about the x-axis the region enclosed by the curve y = sec xand the lines x =, y =, and x =? A) B) C) 8 1 D) E) ln + 88ab The region in the first quadrant between the x-axis and the graph of y = e x, x = 1and the coordinate axes. If the region is rotated around the y-axis, the volume of the solid that is generated is represented by A) B) C) D) 1 xe 1 x e 1 4 x e e y x dx dx dx ln y dy E) e 4 ln y dy 88bc 9 The region R in the first quadrant is enclosed by the lines x = and y = 5 and the graph of y = x + 1. the volume of the solid generated when R is revolved about the y-axis is A) 6 B) 8 C) D) 16 E) 15 Mike Koehler 8-8 Area Volume

9 88bc 6 Let R be the region between the graphs of y = 1 and y = sin x from x = to x =. The volume of the solid obtained by revolving R about the x-axis is given by A) xsin x dx B) x cos x dx D) sin x dx E) ( ) C) ( ) 1 sin x dx 1 sin x dx 85ab 45 The region enclosed by the graph of y = x, the line x =, and the x-axis is revolved about the y-axis. The volume of the solid generated is A) 8 B) 5 C) 16 D) 4 E) 8 85bc 5 The region in the first quadrant between the x-axis and the graph of The volume of the resulting solid of revolution is given by 6 A) ( ) 6 B) ( 6 ) 6 C) ( 6 ) 6x x dx x x x dx x x x dx 6 D) ( ) + 9 y dy 9 E) ( ) + 9 y dy y x x = 6 is rotated around the y-axis. 7ab 5 The region in the first quadrant bounded by the graph of y = sec x, x =, and the x-axis is rotated about the 4 x-axis. What is the volume of the solid generated? 8 A) B) 1 C) D) E) 4 Mike Koehler 8-9 Area Volume

10 Mike Koehler 8-1 Area Volume

11 Free Response 1 AB 6B AB1 x x x Let f be the function given by f( x) = + cos x. 4 Let R be the region in the second quadrant bounded by the graph of f, and let S be the region bounded by the graph of f and the line, the line tangent to the graph of f at x =, as shown in the graph on the right. a) Find the area of R. b) Find the volume of the solid generated when R is revolved about the horizontal line y =. c) Write but do not evaluate an integral expression that can be used to find the area of S. 4B AB1 Let R be the region enclosed by the graph of y = x 1, the vertical line x = 1, and the x-axis. a) Find the area of R. b) Find the volume of the solid generated when R is revolved about the horizontal line y =. c) Find the volume of the solid generated when R is revolved about the vertical line x = 1. B AB1 Let f be the function given by f( x) = 4x x, and let be the line y = 18 x, where is tangent to the graph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f, the line, and the x-axis, as show on the right. a) Show that is tangent to the graph of y = f( x) at the point x =. b) Find the area of S. c) Find the volume of the solid generated when R is revolved about the x-axis. Mike Koehler 8-11 Area Volume

12 B AB1 Let R be the region bounded by the y-axis and the graphs x of y = and y = 4 x, as shown in the figure on the 1 + x right. a) Find the area of R. b) Find the volume of the solid generated when R is revolved about the x-axis. c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid. 1 AB1 Let R and S be the regions in the first quadrant shown in the figure on the right. The region R is bounded by the x-axis and the graphs of y = x and y = tan( x). The region S is bounded by the y-axis and the graphs of y = x and y = tan( x). a) Find the area of R. b) Find the area of S. c) Find the volume of the solid generated when S is revolved about the x-axis. AB1 Let R be the shaded region in the first quadrant enclosed by the x graphs of y = e, y = 1 cos( x), and the y-axis, as shown in the figure on the right. a) Find the area of the region R. b) Find the volume of the solid generated when the region R is revolved about the x-axis. c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid. Mike Koehler 8-1 Area Volume

13 1999 AB The shaded region R is bounded by the graph of y = x and the line y = 4, as shown in the figure on the right. a) Find the area of R. b) Find the volume of the solid generated by revolving R about the x-axis. c) There exists a number k, k > 4, such that when R is revolved about the line y = k, the resulting solid has the sam volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression tha can be used to find the value of k AB1 Let R be the region bounded by the x-axis, the graph of y = x and the line x = 4. a) Find the area of the region R. b) Find the value of h such that the vertical line x = hdivides the region R into two regions of equal area. c) Find the volume of the solid generated when R is revolved about the x-axis. d) The vertical line x = k divides the region R into two regions such that when these two regions are revolved about the x-axis, they generate solids with equal volumes. Find the value of k. Mike Koehler 8-1 Area Volume

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15 Textbook Problems Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l1 Section Questions R Handouts Mike Koehler 8-15 Area Volume

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17 AP Calculus Chapter 7 Section Curve y = x or y = x or x = y. The curve and the lines y = and x = 4 forms region OAC. B C (,8) (4,8) The curve and the lines y = 8 and x = forms region OBC. Find the volumes of the following solids of revolution. 1. OAC rotated about the x-axis. (disk). OAC rotated about the y-axis. (washer and shell). OAC rotated about BC. (washer and shell) O A (4,) 4. OAC rotated about AC. (disk and shell) 5. OAC rotated about y = 1. (washer and shell) 6. OAC rotated about x = 6. (washer and shell) 7. OBC rotated about the x-axis. (washer and shell) 8. OBC rotated about the y-axis. (disk and shell) 9. OBC rotated about BC. (disk and shell) 1. OBC rotated about AC. (washer and shell) 11. OBC rotated about y = -. (washer and shell) 1. OBC rotated about x = 6. (washer and shell) Find the volumes of the following solids with cross sectional areas of OBC as stated o - 45 o - 9 o right triangle with hypotenuse from y-axis to curve. 14. o - 6 o - 9 o right triangle with shorter leg from y-axis to curve. 15. Square with side from y-axis to curve. 16. Square with diagonal from y-axis to curve. 17. Semi-circle with diameter from y-axis to curve. Mike Koehler 8-17 Area Volume

18 Answers Mike Koehler 8-18 Area Volume

19 AP Calculus Chapter 7 Section 1. The region R in the figure on the right is bounded by the graphs of y = kx (where k 1), y = sin( x), and the line x =. The region S is bounded by the graphs of y = sin( x), the line x =, and the x-axis.. 1. R S / a) If k =, find the area of R. b) Find the value of k so that the area of R is. c) If k =, write an integral expression for the volume of the solid generated by revolving R about the x-axis. Find the value of this expression. (calculator) d) Write an equation that could be used to find the value of k so that the area of region R is equal to the area of region S. Solve for k.. The region R shown on the right is bounded by the x x e graphs of y = + 1, y =, and the y-axis R 1... a) Find the area of R. b) The area of R is divided in half by the vertical line x = k. Find the value of k, for < k <. c) Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the x-axis are squares. d) The volume of the solid described in part (c) is divided in half by a plane perpendicular to the x-axis containing the line x = j. Write an equation that could be used to find the value of j. Solve for j. (calculator) Mike Koehler 8-19 Area Volume

20 n. Let be the line tangent to the graph of y = x at the point (1,1), where n > 1, as shown on the right. y = x n a) Find 1 n x dx in terms of n. b) 1 Let T be the triangular region bounded by, the x-axis, and the line x = 1. Show that the area oft is n. n c) Let S be the region bounded by the graph of y = x, the line, and the x-axis. Express the area of S in terms of n and determine the value of n that minimizes the area of S. Mike Koehler 8 - Area Volume

21 AP Calculus Chapter 7 Section Answers 1 a) b) c) d) A = ( x sin( x) ) dx = A = ( kx sin( x) ) dx = k = (( ) ( ) ) V = x sin( x) dx = 4.8 = ( ) A = kx sin( x) dx = 1 k = The curves intersect at a =.6149 a) x a x e A = 1 dx.14 + = 6 b) x k x e.14 A = 1 dx 1.17 k = = = 6 c) x a x e V = + 1 dx =.69 6 d) x j x e.69 V = + 1 dx = = j = a) 1 n +1 b) Show the work. c) 1 1 Area n+ 1 n Maximum area at n 1 Mike Koehler 8-1 Area Volume