AP CALCULUS AB 2003 SCORING GUIDELINES

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1 CORING GUIDELINE Question Let R be the shaded region bounded by the graphs of y the vertical line =, as shown in the figure above. = and y = e and (b) Find the volume of the solid generated when R is revolved about the horizontal line y =. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the -ais is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. Point of intersection e = at (T, ) = (.874,.48864) : Correct limits in an integral in (a), (b), or (c) (a) Area = ( ) T e d =.44 or.44 : : integrand (b) Volume = ( ) ( ) T e d =.45 or.4 or.44 : : integrand < > reversal < > error with constant < > omits in one radius < > other errors (c) Length = e Height = 5( e ) e d =.554 Volume = 5( ) T : : integrand < > incorrect but has e as a factor Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com. 7

2 4 CORING GUIDELINE (Form B) Question Let R be the region enclosed by the graph of y =, the vertical line =, and the -ais. (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 =. (a) Area = d = 8 : : limits : integrand (b) Volume = π 9 ( ) =.57 or.58 d : : limits and constant : integrand (c) Volume = π ( ( y + )) = 47.5 dy : : limits and constant : integrand Copyright 4 by College Entrance Eamination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 8

3 4 CORING GUIDELINE Question Let f and g be the functions given by f ( ) = ( ) and g( ) = ( ) for. The graphs of f and g are shown in the figure above. (a) Find the area of the shaded region enclosed by the graphs of f and g. (b) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved about the horizontal line y =. (c) Let h be the function given by h = k( ) for. For each k >, the region (not shown) enclosed by the graphs of h and g is the base of a solid with square cross sections perpendicular to the -ais. There is a value of k for which the volume of this solid is equal to 5. Write, but do not solve, an equation involving an integral epression that could be used to find the value of k. (a) Area = ( f( ) g( ) = ( ( ) ( ) ) d =. : { : integral (b) Volume = π ( g( ) ) ( f( ) = π (( ( ) ) ( ( ) ) 4 : = 6.79 : limits and constant : integrand each error Note: if integral not of form b c ( R ( ) r ( ) a (c) Volume = ( h g ) d ( k( ) ( ) ) d = 5 : { : integrand Copyright 4 by College Entrance Eamination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 9

4 5 CORING GUIDELINE (Form B) Question Let f and g be the functions given by f ( ) = + sin( ) and g( ) = e. Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure above. (b) Find the volume of the solid generated when R is revolved about the -ais. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the -ais are semicircles with diameters etending from y = f( ) to y = g( ). Find the volume of this solid. The graphs of f and g intersect in the first quadrant at (, T ) = (.569, ). : correct limits in an integral in (a), (b), or (c) (a) Area = ( f( ) g( ) = ( + sin( ) e =.49 : integrand : (b) Volume = π ( f( ) ) ( g( ) = π (( + sin( ) ) ( e ) = 4.66 or 4.67 : integrand each error Note: if integral not of form : b c ( R ( ) r ( ) a (c) Volume π f( ) g( ) = π + sin( ) e = =.77 or.78 d d : integrand : Copyright 5 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents).

5 5 CORING GUIDELINE Question Let f and g be the functions given by f ( ) = + sin ( π ) and g = 4. Let 4 R be the shaded region in the first quadrant enclosed by the y-ais and the graphs of f and g, and let be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (b) Find the area of. (c) Find the volume of the solid generated when is revolved about the horizontal line y =. f ( ) = g( ) when + sin ( π ) = 4. 4 f and g intersect when =.788 and when =. Let a =.788. a (a) ( g ( ) f ( )) d =.64 or.65 : : limits : integrand (b) ( f ( ) g ( )) d =.4 a : : limits : integrand (c) π ( f( ) + ) ( g( ) + ) d = or a : { : integrand : limits, constant, and answer Copyright 5 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents).

6 7 CORING GUIDELINE Question Let R be the region in the first and second quadrants bounded above by the graph of below by the horizontal line y =. (b) Find the volume of the solid generated when R is rotated about the -ais. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the -ais are semicircles. Find the volume of this solid. y = and + + = when =± : correct limits in an integral in (a), (b), or (c) (a) Area = d = 7.96 or : { : integrand (b) Volume = π d = : { : integrand (c) Volume π = + π = d = + d : { : integrand 7 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 5

7 8 CORING GUIDELINE (Form B) Question Let R be the region in the first quadrant bounded by the graphs of y = and y =. (b) Find the volume of the solid generated when R is rotated about the vertical line =. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-ais are squares. Find the volume of this solid. The graphs of y (, ) and ( 9, ). = and y = intersect at the points 9 (a) ( ) d = 4.5 OR y y dy = 4.5 : : limits : integrand (b) π ( + ) ( + ) y y dy 4 : 7π = =.6 or.6 5 : constant and limits : integrand (c) ( y y ) dy = : { : integrand 8. : limits and answer 8 The College Board. All rights reserved. Visit the College Board on the Web: 6

8 CORING GUIDELINE Question Let R be the region bounded by the graphs of y = sin( π ) and above. y = 4, as shown in the figure (b) The horizontal line y = splits the region R into two parts.

8 8 CORING GUIDELINE Question Let R be the region bounded by the graphs of y = sin( π ) and above. y = 4, as shown in the figure (b) The horizontal line y = splits the region R into two parts. Write, but do not evaluate, an integral epression for the area of the part of R that is below this horizontal line. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the -ais is a square. Find the volume of this solid. (d) The region R models the surface of a small pond. At all points in R at a distance from the y-ais, the depth of the water is given by h =. Find the volume of water in the pond. (a) sin ( π ) = 4 at = and = Area ( ) = sin π 4 d = 4 : : limits : integrand (b) 4 = at r = and s =.6759 s The area of the stated region is ( ( 4 )) r d : { : limits : integrand (c) Volume = ( sin ( π ) ( 4 )) d = : { : integrand (d) Volume = ( )( sin( π ) ( 4 )) d = 8.69 or 8.7 : { : integrand 8 The College Board. All rights reserved. Visit the College Board on the Web: 7

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B) AP CALCULUS AB 004 SCORING GUIDELINES (Form B) Question 4 The figure above shows the graph of f, the derivative of the function f, on the closed interval 1 x 5. The graph of f has horizontal tangent lines