Area - Volume Problems 1. The area of the region enclosed by the graph of y = X2 + 1 and the line y = 5 is 14 a. - 3 b. 16 - 3 28 c. - 3 d. 32 3 e. 81r 2. What is the area of the region between the graphs of y = X2 and y = -x from x = 0 to x = 2 ? 2 i a. 3 b. 8 - .~ 3 ,. c. 4 d. 14 - 3 16 e. 3 3. Let R be the region enclosed by the graph of y = 1 + ht( cos 4 x) the x-axis, and the lines x = -~ and x = ~. The closest integer approximation of the area of R is a. 0 b. I c. 2 d. 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. 0.127 b. 0.385 c. 0.400 d. 0.600 e. 0.947 5. If 0 ~ k <~ and the area under the curve y = cosx from x = k to x =~ is 0.1, then k = a. 1.471 b. 1.414 c. 1.277 d. 1.120 e. 0.436 ._- ... --- ------
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Area - Volume Problems
1. The area of the region enclosed by the graph of y = X2 + 1 and the line y = 5 is
14a. -3
b.16-328c. -3
d. 323
e. 81r
2. What is the area of the region between the graphs of y = X2 and y = -x from x = 0 to x = 2 ?
2 ia.3
b. 8- .~3 ,.
c. 4
d. 14-316e.3
3. Let R be the region enclosed by the graph of y = 1+ ht( cos4x) the x-axis, and the lines x = -~ and x = ~. The
closest integer approximation of the area of R is
a. 0b. Ic. 2d. 3e. 4
4. What is the area of the region in the first quadrant enclosed by the graphs ofy =cos x, y =x, andthe y-axis?
a. 0.127b. 0.385c. 0.400d. 0.600e. 0.947
5. If 0 ~ k < ~ and the area under the curve y = cosx from x = k to x = ~ is 0.1, then k =
a. 1.471b. 1.414c. 1.277d. 1.120e. 0.436
._- ... --- ------
6. If the region enclosed by the y-axis, the line y =2, and the curvey = fx is revolved about they -axis, the volume of the solid generated is
32na. 5
b. 16n3
16nc. 5
d. 8n3
e. 'It
7. The base ofa solid S is the region enclosed by the graph ofy = ~, the line x = e. and thex-axis. If the cross sections of S perpendicular to the z-axis are squares, then the volume of S is
Ia. -2
b. 2- -j3c. I .;id. 2
e. ~ (e3 -I)
8. The base ofa solid is the region in the first quadrant enclosed by the graph ofy = 2 - X2 and thecoordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, the volumeof the solid is given by .
f 2a. n (2-y) dy
b. f (2-y)dy
c. n{./2 (2 - X2 rdx
d. r./2 ( 2r2-x dx
e. r./2 (2-x2)dx
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 crosssection perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid?
a. 1.333b. 1.067c. 0.577d. 0.462e. 0.267
y
4
--~--~------~~.x810.
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, asshown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is thevolume of the solid?
a. 12.566b. 14.661c. 16.755 Id. 67.021e. 134.041
.;i]
11. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = tan -Ix, the horizontalline y =3, and the vertical line x = I. For this solid, each cross section perpendicular to the x-axis is a square. Whatis the volume of the solid?
a. 2.561b. 6.612c. 8.046d. 8.755e. 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 = 2x and y = x2, as shown in thefigure above.
(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 to the x-axis
has area 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 to the y-axis are squares.Write, but do not evaluate, an integral expression for the volume of the solid.
y
o-I
--~~--------~---------'r-~x
-2LAklA.l~+O~2-00'6 )<~ 4,~q1. Let R be the region bounded by the graphs of y = sin(Jrx) and y = x3 - 4x, as shown in the figure above.
-3
(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 integralexpression 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, thedepth 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 (L.
~b y:~ L\.5"Lf1. 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 generatedwhen R is rotated about the y-axis. ;~
.~,.
2009 API!)CALCULUS AB FREE-RESPONSE QUESTIONS (Form B)
No calculator.
y
1
2
--~--~~--~----~-----+--~xo 1 2 3 4
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 rotatedabout the horizontal line y = 2.
©
;------ -- -------
Ap® CALCULUS AB2009 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.
(a) Find the area of R.(b) The region R is the base ofa solid. For this solid, at each x the cross
section perpendicular to the x-axis has area 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 to the y-axis are squares. Write, but do not evaluate, anintegral expression for the volume of the solid.
4
3
2
2
(a) Area = J:(2X - x2) dx
= x2 _ .!.x31 x=2
3 x=Q
(b) Volume = L\in( ~ x) dx
2 (1r)1 x=Z= --cos -x1r 2 x=O
4=
(4 2(c) Volume = J
o(& - i) dy
{
I: integrand3: 1: antiderivative
1 : answer
{
I: integrand3: 1: antiderivative
1 : answer .
{2 : integrand3:1 : limits
© 2009 The College Board. All rights reserved.Visit the College Board on th84Web: www.collegeboard.com.
Ap® CALCULUS AB2008 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 figureabove.
(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 to the x-axis is asquare. 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 hex) = 3 - x. Find the volume of water in the pond.
(a) sin(1l"x) = x3 - 4x at x = 0 and x = 2
Area = 5:(sin (1l"X) - (x3- 4x )) dx = 4
(b) x3 - 4x = -2 at r = 0.5391889 and s = 1.6751309
The area of the stated region is s: (-2 - (x3- 4x)) dx
f2 2
(c) Volume = o (sin(1l"x)-(x3 -4x)) dx=9.978
(d) Volume = 5:(3 - x) (sin (zr») - (x3- 4x)) dx = 8.369 or 8.370
{
I: limits3: 1: integrand
1 : answer
{I: limits .2'. 1: integrand
2 : { I : integrand1 : answer
{I: integrand2:I : answer
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Ap® CALCULUS AB2006 SCORING GUIDELINES
Let R be the shaded region bounded by the graph of y = In x and the liney = 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 thevolume of the solid generated when R is rotated about the y-axis.
Question 1
--+--+-~:....----x
In(x) = x - 2 when x = 0.15859 and 3.14619.Let S = 0.15859 and T = 3.14619
(a) Area of R= S;(ln(X)-(X-2))dx = 1.949
(b) Volume = 1r S; ((In (x) + 3)2 - (x - 2 + 3)2 ) dx
= 34.198 or 34.199
fT-2( 2)(c) Volume = 1r S-2 (y + 2)2 - (eY) dy
{
I: integrand3: 1: limits
1 : answer
{2 : integrand
3 :1 : limits, constant, and answer
3 : { 2 : integrand1 : limits and constant
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2
Ap® CALCULUS AB2009 SCORING GUIDELINES (Form B)
(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-----!-_xvolume of the solid generated when R is rotated about 0the horizontal line y = 2.
Question 4
Let R be the region bounded by the graphs of y = -IX and Y
y = 1, as shown in the figure above.2
1 3 i 42
4 21X=4 4(a) Area = ( (-IX -~) dx = !:"x3/2 - ~ =
Jo 2 3 4 x=o 3
2 5/2 31X=4=~_~+~ 82 5 12 x=o 15
(c) Volume = "f((2 - ~)' - (2 - .IX)' ) dx
{
I: integrand3: 1: antiderivative
1 : answer
{
I: integrand3: 1: antiderivative
1 : answer
{I: limits and constant
3:2 : integrand
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