2e AP Exam Prep Solutions Ch 06 WEB
AP ? Exam Practice Questions for Chapter 6
1
AP? Exam Practice Questions for Chapter 6
y
1.
y=e
3
2
1
y = e 3x
?1
1
2
x
3
?1
e3 x = e
3x = 1
x =
1
3
A =
3x
?0 (e ? e ) dx
13
13
1 ?
?
= ?ex ? e3 x ?
3 ?0
?
(
= e?
) (
1
3
? 13 e ? 0 ?
1
3
)
1
3
=
So, the answer is A.
2.
y = x 3 ? 7x 2 + 12x + 4
y
16
14
12
10
8
6
4
y = 2x + 4
(5, 14)
(2, 8)
(0, 4)
?1
1
2
3
4
5
6
x
x3 ? 7 x 2 + 12 x + 4 = 2 x + 4
x3 ? 7 x 2 + 10 x = 0
x( x 2 ? 7 x + 10) = 0
x( x ? 5)( x ? 2) = 0
x = 0, 2, 5
A =
=
?0 ??( x
2
?0 ( x
2
3
? 7 x 2 + 12 x + 4) ? ( 2 x + 4)?? dx +
3
? 7 x 2 + 10 x) dx +
?2 (? x
5
2
3
= ?? 14 x 4 ? 73 x3 + 5 x 2 ?? + ??? 14 x 4 +
0
(
= 4?
=
56
3
) (
+ 20 + ? 625
+
4
875
3
?2 ??(2 x + 4) ? ( x
5
3
? 7 x 2 + 12 x + 4)?? dx
+ 7 x 2 ? 10 x) dx
7 x3
3
5
? 5 x 2 ??
2
) (
? 125 ? ? 4 +
56
3
)
? 20
253
12
So, the answer is C.
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AP ? Exam Practice Questions for Chapter 6
2
3. A =
¦Ð 4
?0 (4 cos x ? 4 sin x) dx
6.
y = ln (sec x)
dy
1
=
? sec x tan x
sec x
dx
¦Ð 4
= [4 sin x + 4 cos x] 0
= tan x
? ? 2?
? 2 ??
= ?4??
?? + 4??
??? ? ??0 + 4(1)??
2
?
? 2 ???
?? ?
?0
=
?0
=
?0
= 4 2 ?4
(
)
= 4
2 ?1
So, the answer is A.
1
2
A =
7.
6
= [2 x + 4 cos x] 0
?¦Ð
?
= ? + 2 3 ? ? ( 0 + 4)
?3
?
3
+ 2 3 ?4
8.
(0, 4)
?2
?1
1
?2
6
?1
=
?1
?3
=
?3
x
2
( 27 ? 8)
3
38
=
3
y
3
y = ?1, 4
4
2
= ??? 13 y 3 +
(
125
6
So, the answer is D.
1
?1
2
x
3
?1
4
+ 4 y??
?1
) ( 13 + 23 ? 4)
= ? 64
+ 24 + 16 ?
3
=
? 4 y )?? dy
+ 3 y + 4) dy
3 y2
2
1 + x dx
=
9.
2
??1 ??(4 ? y) ? ( y
2
So, the answer is B.
? 4)( y + 1) = 0
4
8
1 + ( x1 2 ) dx
8
y2 ? 3y ? 4 = 0
??1 (? y
1 + 4e x dx
3 2?
?2
= ? (1 + x) ?
3
?
?3
(5, ? 1)
y2 ? 4 y = 4 ? y
=
8
s =
y = ?x + 4 ? x = 4 ? y
A =
2
2 32
x
3
y =
Write both equations in terms of y and find the points
of intersection.
(y
4
1 + ( 2e0.5 x ) dx
dy
= x1 2
dx
y
?4
4
s =
So, the answer is A.
So, the answer is B.
5.
sec x dx
= 2e0.5 x
?0 (2 ? 4 sin x) dx
¦Ð
¦Ð 4
sec 2 x dx
y = 4e0.5 x
¦Ð 6
¦Ð 6
=
¦Ð 4
dy
= 0.5( 4e0.5 x )
dx
¦Ð
x =
1 + tan 2 x dx
So, the answer is B.
4. 4 sin x = 2
sin x =
¦Ð 4
s =
V =
1
?0 ??2 ? sin
?1
2
x?? dx
¡Ö 2.184
So, the answer is C.
? Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
AP ? Exam Practice Questions for Chapter 6
3
y
10. (a)
12
8
Reminders: Use your calculator to find the point of
intersection of the two graphs (no work needed for
this).
6
4
k
(0, 1)
?1
1
?2
2
3
4
x
5
Find the point of intersection of the graphs.
In this intermediate step, round the x-coordinate of
the intersection point to more than three decimal
places to use in upcoming integrals.
10 = 1 + 6 x3 2
9 = 6 x3 2
3
= x3 2
2
?3?
? ?
?2?
23
= x
x ¡Ö 1.3103707
32
?0 ??10 ? (1 + 6 x )?? dx = ?k
k
1.3103707
?10 ? (1 + 6 x3 2 )? dx
?
?
or
y = 1 + 6 x3 2
y ? 1 = 6 x3 2
y ?1
= x3 2
6
? y ? 1?
x = ?
?
? 6 ?
y ? 1?
?
?
? 6 ?
10 ?
?1
23
23
dy = 2 ?
1+ 6 k 3 2
0
? y ? 1?
?
?
? 6 ?
23
dy
y = 1 + 6 x3 2
(b)
? 3?
y¡ä = 6? ? x1 2 = 9 x1 2
? 2?
1 + ( y¡ä) = 1 + (9 x
2
s =
1
?0
12
)
2
= 1 + 81x
1 + 81x dx
=
1 1
(1 ? 81x)1 2 (81) dx
81 ?0
=
1 ?2
(1 + 81x)3 2 ??
81 ?? 3
?0
=
2
(823 2 ? 1) ¡Ö 6.103
243
Reminders: Be sure to write down the appropriate
definite integral before numerically approximating
it on your calculator.
Be sure to round the answer to at least three decimal
places to receive credit on the exam.
1
(c) A = w ? h
= ??10 ? (1 + 6 x 3 2 )?? ? 3 ??10 ? (1 + 6 x 3 2 )??
= 3(9 ? 6 x 3 2 )
V = 3?
1.3103707
0
2
(9 ? 6 x 3 2 )
2
dx ¡Ö 143.289
Reminders: Be sure to write down the appropriate
definite integral before numerically approximating
it on your calculator.
Be sure to round the answer to at least three decimal
places to receive credit on the exam.
? Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
4
AP ? Exam Practice Questions for Chapter 6
y
11. (a)
3
2
1
?3 ?2 ?1
?1
2
3
x
4
Reminders: Use your calculator to find the points of
intersection of the two graphs (no work needed for
this).
?2
?3
In this intermediate step, round the coordinates of
the intersection points to more than three decimal
places to use in upcoming integrals.
Rewrite the equations in terms of y.
y = ln x ? x = e
y
y = 2x ? 3 ? x =
1
3
y +
2
2
The graphs intersect at y ¡Ö ? 2.888703
and y ¡Ö 0.58307388.
A =
0.58307388
?? 2.888703
?? 1
3?
y?
?? 2 y + 2 ? ? e ? dy
?
?
?
?
0.58307388
3
?1
?
= ? y2 + y ? e y ?
2
?4
? ? 2.888703
¡Ö ( ? 0.832) ? ( ? 2.303)
= 1.471
(b) V = ¦Ð
? 0.05564832 ( ??ln( x) + 3??
1.7915369
2
? ??( 2 x ? 3) + 3??
2
) dx
¡Ö 18.783
Reminders: Be sure to write down the appropriate
definite integral before numerically approximating
it on your calculator.
Be sure to round the answer to at least three decimal
places to receive credit on the exam.
(c) ¦Ð
0.58307388 ??
??2.888703
??
???
2
2?
y + 3?
y
? ? (e ) ? dy
2 ?
??
Notes: To use disks (washers), set up this integral in
terms of y (using horizontal elements in R).
Setting up the volume integral in terms of x (using
vertical elements in R) requires the shell method.
? Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
AP ? Exam Practice Questions for Chapter 6
5
y
12. (a)
4
3
2
1
?4 ?3 ?2 ?1
2
3
x
4
Note: Because R is horizontally simple, it is strategic
to set up this area integral in terms of y (using horizontal
elements in R). Setting up the area integral in terms of x
(using vertical elements in R) requires the sum of
multiple definite integrals because R is not vertically
simple.
?2
?3
?4
Find the points of intersection of the graphs
y = x2 ? 1 ? x = ¡À
y +1
Reminders: Use your calculator to find the points of
intersection of the two graphs (no work needed for this).
y + 1 = y2
y + 1 = y4
0 = y4 ? y ? 1
y ¡Ö ? 0.724492, 1.2207441
A =
??0.724492 (
)
1.2207441
Be sure to write down the appropriate definite integral
before numerically approximating it on your calculator.
y + 1 ? y 2 dy
32
= ? 23 ( y + 1) ?
?
1.2207441
1 y3 ?
3 ? ? 0.724492
Be sure to round the answer to at least three decimal
places to receive credit on the exam.
¡Ö 1.377
(b) V = ¦Ð
?
2
? ?0.724492 ??(2 ? y )
1.2207441
2
In this intermediate step, round the coordinates of the
intersection points to more than three decimal places to
use in upcoming integrals.
(
)
2
y + 1 ?? dy
?
? 2 ?
¡Ö 11.501
Notes: To use disk (washers), set up this integral in terms
of y (using horizontal elements in R).
Setting up this volume integral in terms of x (using
vertical elements in R), via the shell method, requires
the sum of multiple definite integrals because R is
not vertically simple.
Reminders: Be sure to write down the appropriate
definite integral before numerically approximating it
on your calculator.
Be sure to round the answer to at least three decimal
places to receive credit on the exam.
(c) 2¦Ð
¦Ð
? ?0.724492 ( y + 1)(
)
1.2207441
0.5248886
?0
(
?
??
+¦Ð
y + 1 ? y 2 dy or
) ? (?
x +1
2
?
? 0.524886 ??(
1.4902161
)
2
x + 1 ?? dx
?
) ? ( x ) ?? dx
x +1
2
2
2?
Notes: Because R is horizontally simple, it is strategic to
set up the shell method here (using horizontal elements
in R) in order to set up a single integral.
Setting up this volume integral in terms of x (using
vertical elements in R), via the disk/washer method,
requires the sum of multiple definite integrals because
R is not vertically simple.
? Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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