Please give solutions to the (application of integration to geometry) questions below. Step-by-step explanation is much appreciated.

VOLUME In Questions 18-24 the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated. 322 AP Calculus 1 . . 4L y= xz 22. y = x2 and y = 4; about the line y = 4. 256TC 512TC (A) (B) 256TC (C) (D) 512n (E) 5 3 15 5 15 23. : An arch of y = sin x and the x-axis; about the x-axis. () (7 - ) ( B ) (C) (D) T2 (E) I(J- 1) 24. A trapezoid with vertices at (2, 0), (2, 2), (4, 0), and (4, 4); about the x-axis. (A) 56TC (B) 1281 (C) 92 TC 3 3 3 (D 112n (E) 20n 3 = 41 " 9 - 4 un - jo The base of a solid is the region bounded by the parabola x? = 8y and the line y = 4, and each plane section perpendicular to the y-axis is an equilateral triangle. The volume of the solid is (A) 6413 (B) 64V3 (C) 32V3 3 321/3 (D) 32 (E 3 27. The base of a solid is the region bounded by y = ed, the x-axis, the y-axis, and the line x = 1. Each cross section perpendicular to the x-axis is a square. The volume of the solid is A (B) e - 1 (C) 1- (D) -1 2 (E) (1 -2 )calculator. Part B. Directions: Some of the following questions require the use of a graphing AREA boundaries are given. In Questions 43-47, choose the alternative that gives the area of the region whose 43. The area bounded by the parabola y = 2 - x3 and the line y = x - 4 is given by 2- X2 = X - 4 L (6-x-x7) dx (B) [ (2+x + x ) dx (0) 1 16-x-27) dx (D). 2 (2 - x7 ) dox + / (4 - 20 ) dox ( ) [, [( + 4) -12- y]dy BC ONLY (0,1) (1,0) 44. The area enclosed by the hypocycloid with parametric equations x = cos' t and y = sin' t as shown in the above diagram is (A) 3 sin' tcost tat (B ) 4 ) sin' tat ( C ) 4 sins t dt (D) 12 sin* t cost t at ( E) 6 sin' t cos ' t dt 45. Suppose the following is a table of coordinates for y = f(x), given that f is continu- ous on [1, 5]: * 1 2 3 4 5 y 1.62 4.15 7.5 9.0 12.13 If a trapezoid sum in used, with n = 4, then the area under the curve, from x = 1 to x = 5, is equal, to two decimal places, to (A) 6.88 (B) 13.76 (C) 20.30 (D) 25.73 (E) 27.53 VOLUME 2 In Questions 48-54 the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated. 52. The curve with parametric equations x = tan 0, y = cos? 0, and the lines x = 0, BC ONLY x = 1, and y = 0; about the x-axis. (A) I COs* 0. de Jo (B) it Cos2 0 sin e de (C) i Cos ' 0 de (D) I COS2 0 de (E) I Cos* e de 54. If the curves of f(x) and g(x) intersect for x = a and x = b and if f(x) > g(x) > 0 for all x on (a, b), then the volume obtained when the region bounded by the curves is 'S rotated about the x-axis is equal to 56. The length of the arc of the parabola 4x = y cut off by the line x = 2 is given by the (A) I f' ( x) dx - 82 (x) dx integral (A) Vitlax (B ) 1 4ty dy ( c) [ VI+ x dx (B) x [f(x) - 8(x)12 dx IVAtydy ( ) [ Vatydy (C) 21 x [f (xx) - 8()] doc 57. The length of x = e' cost, y = e' sin t from t = 2 to t = 3 is equal to (D) * [f ( x ) - 8"(20) ] dox (A) Vie' Ve? - 1 ( B ) V2(@ 3 - e ? ) ( C ) 2 ( e ' - e ? ) (E) 21 157 ( x) -82 (xx) ] dox (D) e'(cos 3 + sin 3) - e-(cos 2 + sin 2) ( E ) 2 Ve - e +