AP® CALCULUS AB 2003 SCORING GUIDELINES

Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.

2

Question 1

Let R be the shaded region bounded by the graphs of y x= and 3xy e= and

the vertical line 1,x = 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 horizontal

line 1.y = (c) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis 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 3xe x= at (T, S) = (0.238734, 0.488604)

1: Correct limits in an integral in

(a), (b), or (c)

(a) Area = ( )1 3xTx e dx

= 0.442 or 0.443

2 : 1 : integrand

1 : answer

(b) Volume = ( ) ( )( )1 2231 1xT

e x dx

= 0.453 or 1.423 or 1.424

3 :

2 : integrand

1 reversal

1 error with constant

1 omits 1 in one radius

< 2 other errors

1 : answer

< >

< >

< >

>

(c) Length = 3xx e

Height = ( )35 xx e

Volume = ( )1 235 xT

x e dx = 1.554

3 : 3

2 : integrand

< 1 > incorrect but has

as a factor

1 : answer

xx e

7

AP® CALCULUS AB 2004 SCORING GUIDELINES (Form B)

Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).

2

Question 1

Let R be the region enclosed by the graph of 1,y x= − the vertical line 10,x = 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 3.y = (c) Find the volume of the solid generated when R is revolved about the vertical line 10.x =

(a) Area 10

11 18x dx= − =

3 :

1 : limits

1 : integrand

1 : answer

(b) Volume ( )( )10 21

9 3 1

212.057 or 212.058

x dxπ= − − −

=

3 :

1 : limits and constant

1 : integrand

1 : answer

(c) Volume ( )( )3 220

10 1

407.150

y dyπ= − +

=

3 :

1 : limits and constant

1 : integrand

1 : answer

8

AP® CALCULUS AB 2004 SCORING GUIDELINES

Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).

3

Question 2

Let f and g be the functions given by ( ) ( )2 1f x x x= − and

( ) ( )3 1g x x x= − for 0 1.x≤ ≤ 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 2.y =

(c) Let h be the function given by ( ) ( )1h x k x x= − for 0 1.x≤ ≤ For each 0,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 x-axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k.

(a) Area ( ) ( )( )

( ) ( )( )

1

01

02 1 3 1 1.133

f x g x dx

x x x x dx

= −

= − − − =

2 : { 1 : integral1 : answer

(b) Volume ( )( ) ( )( )( )1 2 20 2 2g x f x dxπ= − − − ( )( ) ( )( )( )1 2 20 2 3 1 2 2 1

16.179

x x x x dxπ= − − − − −

=

4 :

( ) ( )( )2 2

1 : limits and constant 2 : integrand 1 each error Note: 0 2 if integral not of form

1 : answer

b

ac R x r x dx

−

−

(c) Volume ( ) ( )( )1 20h x g x dx= −

( ) ( )( )1 20

1 3 1 15k x x x x dx− − − =

3 : { 2 : integrand1 : answer

9

AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B)

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

Question 1

Let f and g be the functions given by ( ) ( )1 sin 2f x x= + and ( ) 2.xg x 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.

(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 ( ) ( ), 1.13569, 1.76446 .S T =

1 : correct limits in an integral in (a), (b),

or (c)

(a) Area ( ) ( )( )

( )( )0

2

0= 1 sin 2

0.429

S

S x

f x g x dx

x e dx

= −

+ −

=

2 : 1 : integrand

1 : answer

(b) Volume ( )( ) ( )( )( )( )( ) ( )( )

2 2

0

22 2

0

1 sin 2

4.266 or 4.267

S

Sx

f x g x dx

x e dx

π

π

= −

= + −

=

3 :

( ) ( )( )2 2

2 : integrand

1 each error

Note: 0 2 if integral not of form

1 : answer

b

ac R x r x dx

−

−

(c) Volume ( ) ( )

( )

2

0

22

0

2 2

1 sin 2

2 2

0.077 or 0.078

S

S x

f x g x dx

x e dx

π

π

−=

+ −=

=

3 : 2 : integrand

1 : answer

10

AP® CALCULUS AB 2005 SCORING GUIDELINES

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

Question 1

Let f and g be the functions given by ( ) ( )1 sin4

f x xπ= + and ( ) 4 .xg x −= Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S 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 area of S. (c) Find the volume of the solid generated when S is revolved about the horizontal

line 1.y = −

( ) ( )f x g x= when ( )1 sin 44

xxπ −+ = . f and g intersect when 0.178218x = and when 1.x = Let 0.178218.a =

(a) ( ) ( )( )0

0.064ag x f x dx− = or 0.065

3 :

1 : limits

1 : integrand

1 : answer

(b) ( ) ( )( )1

0.410af x g x dx− =

3 :

1 : limits

1 : integrand

1 : answer

(c) ( )( ) ( )( )( )1 2 21 1 4.558a f x g x dxπ + − + = or 4.559

3 : { 2 : integrand1 : limits, constant, and answer

11

AP® CALCULUS AB 2007 SCORING GUIDELINES

Question 1

© 2007 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).

Let R be the region in the first and second quadrants bounded above by the graph of 220

1y

x=

+ and

below by the horizontal line 2.y =

(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated 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. Find the volume of this solid.

2

20 21 x

=+

when 3x = ±

1 : correct limits in an integral in (a), (b), or (c)

(a) Area 3

23

20 2 37.961 or 37.9621

dxx−

= − =+

2 : { 1 : integrand1 : answer

(b) Volume 3 2

22

3

20 2 1871.1901

dxx

π−

= − =+

3 : { 2 : integrand1 : answer

(c) Volume 3 2

233 2

23

1 20 22 2 1

20 2 174.2688 1

dxx

dxx

π

π−

−

= −+

= − =+

3 : { 2 : integrand1 : answer

15

AP® CALCULUS AB 2008 SCORING GUIDELINES (Form B)

Question 1

© 2008 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

Let R be the region in the first quadrant bounded by the graphs of y x= and .3xy =

(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the vertical line 1.x = − (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are

squares. Find the volume of this solid.

The graphs of y x= and 3xy = intersect at the points

( )0, 0 and ( )9, 3 .

(a) ( )90

4.53xx dx− =

OR

( )3 20 3 4.5y y dy− =

3 : 1 : limits1 : integrand1 : answer

(b) ( ) ( )( )3 22 20

3 1 1

207 130.061 or 130.0625

y y dyπ

π

+ − +

= =

4 : 1 : constant and limits

2 : integrand 1 : answer

(c)

( )3 220 3 8.1y y dy− = 2 : { 1 : integrand1 : limits and answer

16

AP® CALCULUS AB 2008 SCORING GUIDELINES

Question 1

© 2008 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

Let R be the region bounded by the graphs of ( )siny xπ= and 3 4 ,y x x= − as shown in the figure above. (a) Find the area of R. (b) The horizontal line 2y = − splits the region R into two parts. Write, but do not evaluate, an integral

expression 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 x-axis 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 x from the y-axis,

the depth of the water is given by ( ) 3 .h x x= − Find the volume of water in the pond.

(a) ( ) 3sin 4x x xπ = − at 0x = and 2x =

Area ( ) ( )( )2 30 sin 4 4x x x dxπ= − − =

3 : 1 : limits1 : integrand1 : answer

(b) 3 4 2x x− = − at 0.5391889r = and 1.6751309s =

The area of the stated region is ( )( )32 4sr x x dx− − −

2 : { 1 : limits1 : integrand

(c) Volume ( ) ( )( )2 230 sin 4 9.978x x x dxπ= − − = 2 : { 1 : integrand1 : answer

(d)

Volume ( ) ( ) ( )( )2 30 3 sin 4 8.369 or 8.370x x x x dxπ= − − − = 2 : { 1 : integrand1 : answer

17