_ap05_sg_calculus_ab__46570

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Unformatted text preview: Web: www.collegeboard.com. AP Central is the official online home for the AP Program and Pre-AP: apcentral.collegeboard.com. AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 1 Let f and g be the functions given by f ( x ) = 1 + sin ( 2 x ) and g ( x ) = e x 2 . Let R be the shaded region in the first quadrant enclosed by the graphs of f and g as shown in the figure above. (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, the cross sections perpendicular to the x-axis are semicircles with diameters extending from y = f ( x ) to y = g ( x ) . Find the volume of this solid. The graphs of f and g intersect in the first quadrant at ( S , T ) = (1.13569, 1.76446 ) . S ∫0 ( f ( x ) − g ( x ) ) dx S = ∫ (1 + sin ( 2 x ) − e x 2 ) dx 0 (a) Area = 1 : correct limits in an integral in (a), (b), or (c) ⎧ 1 : integrand 2: ⎨ ⎩ 1 : answer = 0.429 (b) Volume = π ∫0 ( ( f ( x ) ) S ⎮ = π⌠ S ⌡0 2 ) − ( g ( x ) )2 dx ((1 + sin ( 2x )) 2 ( − ex ) 22 ⎧ 2 : integrand ⎪ −1 each error ⎪ ⎪ Note: 0 2 if integral not of form 3: ⎨ b ⎪ c ∫ R 2 ( x ) − r 2 ( x ) dx a ⎪ ⎪ 1 : answer ⎩ ) dx ( = 4.266 or 4.267 ) S 2 ⌠ π ⎛ f ( x) − g( x) ⎞ (c) Volume = ⎮ ⎜ ⎟ dx 2 ⎠ ⌡0 2 ⎝ S ⎧ 2 : integrand 3: ⎨ ⎩ 1 : answer 2 ⌠ π ⎛ 1 + sin ( 2 x ) − e x 2 ⎞ =⎮ ⎟ dx 2⎜ 2 ⎠ ⌡0 ⎝ = 0.077 or 0.078 Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 2 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 2 A water tank at Camp Newton holds 1200 gallons of water at time t = 0....
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This note was uploaded on 10/01/2009 for the course OC 9876 taught by Professor Dq during the Spring '09 term at UC Merced.

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