Area - Volume Problems
1. The area of the region enclosed by the graph of y
a.
b.
c.
d.
e.
2.
= X2 + 1 and the line y = 5 is
14
3
16
3
28
3
32
3
81r
-
What is the area of the region between the graphs of y
a.
b.
c.
d.
e.
and y
= -x from x = 0 to x = 2 ?
2
3
8
3
4
14
3
16
3
i
.~
,.
3. Let R be the region enclosed by the graph of y
closest integer approximation
a.
= X2
= 1+ ht( cos x)
4
the x-axis, and the lines x = -~ and x
of the area of R is
0
b.
I
c.
d.
2
3
e.
4
4. What is the area of the region in the first quadrant enclosed by the graphs ofy =cos x, y =x, and
the y-axis?
a.
b.
c.
d.
e.
5.
0.127
0.385
0.400
0.600
0.947
If 0 ~ k < ~ and the area under the curve y
a.
b.
c.
d.
e.
= cosx
1.471
1.414
1.277
1.120
0.436
._-
...
---
------
from x = k to x = ~ is 0.1, then k
=
= ~.
The
6.
If the region enclosed by the y-axis, the line y =2, and the curvey
y -axis, the volume of the solid generated is
a.
b.
c.
d.
e.
7.
is revolved about the
32n
5
16n
3
16n
5
8n
3
'It
= ~,
The base ofa solid S is the region enclosed by the graph ofy
the line x = e. and the
x-axis. If the cross sections of S perpendicular to the z-axis are squares, then the volume of S is
d.
I
2
2
3
I
2
e.
~ (e -I)
a.
b.
c.
8.
= fx
-
-j
.;i
3
The base ofa solid is the region in the first quadrant enclosed by the graph ofy = 2 - X2 and the
coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volume
of the solid is given by
.
a.
b.
f
f
n
(2-y)
2dy
(2-y)dy
{./2 (2-
c.
n
X2
d.
r./2 ( 2-x
2
e.
r./2 (2-x )dx
r
r
dx
dx
2
9. The region bounded by the graph of y = 2x _X2 and the x-axis is the base of a solid. For this solid, each cross
section perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid?
a.
b.
c.
d.
e.
1.333
1.067
0.577
0.462
0.267
y
4
--~--~------~~.x
8
10.
The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x + 2y = 8, as
shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the
volume of the solid?
a.
b.
c.
d.
e.
12.566
14.661
16.755
67.021
134.041
I
.;i]
11. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = tan -I x, the horizontal
line y =3, and the vertical line x = I. For this solid, each cross section perpendicular to the x-axis is a square. What
is the volume of the solid?
a.
b.
c.
d.
e.
2.561
6.612
8.046
8.755
20.773
y
No~~
zccs Ae,
X:: «1.07
4. Let R be the region in the first quadrant enclosed by the graphs of y
figure above.
=
2x and y
=
x2, as shown in the
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, at each x the cross section perpendiculsr
has area
to the x-axis
A(x) = sin(; x). Find the volume of the solid.
(c) Another solid has the same base R. For this solid, the cross sections perpendicular
Write, but do not evaluate, an integral expression for the volume of the solid.
to the y-axis are squares.
y
--~~--------~---------'r-~x
o
-I
-2
LAklA.l~+O~
2-00'6
)<~
-3
4,~q
1. Let R be the region bounded by the graphs of y = sin(Jrx)
and y = x3
-
4x, as shown in the figure above.
(a) Find the area of R.
(b) The horizontal line y
=
-2 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
Find the volume of this solid.
to the x-axis is a square.
(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 h(x) = 3 - x. Find the volume of water in the pond.
-----
----
y
--::+---r--+--- •..
x
C,,-kv..lcd-o
~b
(L.
y:~ L\.5"Lf
1. Let R be the shaded region bounded by the graph of y
=
In x and the line y
=x-
2, as shown above.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is rotated about the horizontal line 'y'= -3.
(c) Write, bu.tdo not evaluate, an int~gral expression that can be used to find the volume ,~fthe solid generated
when R is rotated about the y-axis.
;~
.
2009 API!) CALCULUS
AB FREE-RESPONSE
QUESTIONS
~,.
(Form B)
No calculator.
y
2
1
--~--~~--~----~-----+--~x
o
2
4
1
3
4. Let R be the region bounded by the graphs of y
=
.JX
and y =
I' as shown in the figure above.
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares.
Find the volume of this solid.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the horizontal line y = 2.
©
;
------ --
-------
Ap® CALCULUS AB
2009 SCORING GUIDELINES
Question 4
Let R be the region in the first quadrant enclosed by the graphs of y = 2x and
y
=
y
x2, as shown in the figure above.
4
(a) Find the area of R.
(b) The region
is the base ofa solid. For this solid, at each
R
section perpendicular to the x-axis has area A( x)
x
the cross
= sin ( ~ x).
3
Find the
2
volume of the solid.
(c) Another solid has the same base R. For this solid, the cross sections
perpendicular to the y-axis are squares. Write, but do not evaluate, an
integral expression for the volume of the solid.
2
(a) Area
=
J:(2X -
= x2 _
(b) Volume
.!.x3
x=2
3
x=Q
1
=
L\in(
~ x) dx
=
2
--cos
(1r)1
-x
=
1r
=
2
x=Z
I: integrand
3:
1: antiderivative
{
1 : answer
.
x=O
4
(4
(c) Volume
I: integrand
3:
1: antiderivative
{
1 : answer
x2) dx
2
(&
i)
dy
o
J
3: {
2 : integrand
1 : limits
© 2009 The College Board. All rights reserved.
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Ap® CALCULUS AB
2008 SCORING GUIDELINES
Question 1
y
--~--------~~--------~~-x
-1
-2
-3
Let R be the region bounded by the graphs of y
=
sin(1l"x) and y
=
x3
-
4x, as shown in the figure
above.
(a) Find the area of R.
(b) The horizontal line y = -2 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
square. Find the volume of this solid.
to the x-axis is a
(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 hex) = 3 - x. Find the volume of water in the pond.
(a) sin(1l"x)
Area
(b) x3
-
=
x3
=
5:(
4x
=
-
=
4x at x
sin ( 1l"X)
-2 at r
-
=
3
(x
0 and x
-
4x )) dx
0.5391889
The area of the stated region is
2
(c) Volume =
(d) Volume
=
3
fo (sin(1l"x)-(x
=
=
3:
4
and s
s:
I:
2
=
3
(-2 - (x
-
4x)) dx
3
-
4x)) dx
limits .
2 : { I : integrand
1 : answer
dx=9.978
5:(3 - x) (sin (zr») - (x
1 : answer
2' {
. 1: integrand
2
-4x))
1: integrand
{
I:
1.6751309
limits
= 8.369
or 8.370
I:
2: {
© 2008 The College Board. All rights reserved.
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integrand
I : answer
Ap® CALCULUS AB
2006 SCORING GUIDELINES
Question 1
Let R be the shaded region bounded by the graph of y
y
=
=
In x and the line
x - 2, as shown above.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is rotated about the horizontal
line y = -3.
(c) Write, but do not evaluate, an integral expression that can be used to find the
volume of the solid generated when R is rotated about the y-axis.
In(x)
=x-
Let S
=
(a)
2 when x
= 0.15859
0.15859 and T
Area of R=
=
and 3.14619.
3.14619
=
S;(ln(X)-(X-2))dx
(c)
Volume
S;(
=
1r
=
34.198 or 34.199
(In (x)
fT-2(
Volume = 1r S-2
(y
Visit apcentral.collegeboard.com
+ 3)2 - (x - 2 + 3)2 ) dx
+ 2)2 - (eY)
2)
I: integrand
1: limits
1.949
3:
(b)
--+--+-~:....----x
{
3:{
1 : answer
2 : integrand
1 : limits, constant, and answer
3 : { 2 : integrand
1 : limits and constant
dy
© 2006 The College Board. All rights reserved.
(for AP professionals) and www.collegeboard.com/apstudents
2
(for AP students and parents).
Ap® CALCULUS AB
2009 SCORING GUIDELINES (Form B)
Question 4
Let R be the region bounded by the graphs of y
y
=
= -IX
and
Y
1, as shown in the figure above.
2
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, the
cross sections perpendicular to the x-axis are squares.
Find the volume of this solid.
(c) Write, but do not evaluate, an integral expression for the -!'----f-----,f-----!f-----!-_x
volume of the solid generated when R is rotated about 0
3 i
1
2
4
the horizontal line y = 2.
(a) Area
4
= ( (-IX -~)
Jo
dx
2
=
!:"x3/2
3
- ~
21X=4
4 x=o
=
4
3
I: integrand
3:
1: antiderivative
{
1 : answer
3:
2
=~_~+~
2
(c) Volume
=
5/2
5
31X=4
12 x=o
"f((2 - ~)' - (2 -
I: integrand
1: antiderivative
{
1 : answer
8
15
.IX)' )
dx
I: limits and constant
3: {
2 : integrand
© 2009 The College Board. All rights reserved.
Visit the College Board on theSNeb: www.collegeboard.com.